Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C16 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | C24 |
| commutator, socle | C2 | C24×C16 |
| Frattini, cosocle | C16 | C25 |
| prime radical | #585 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 128 points:
(1,2,5,11,6,12,24,39,7,13,25,40,27,42,61,81,8,14,26,41,28,43,62,82,29,44,63,83,64,84,99,114)(3,9,17,32,18,33,52,72,19,34,53,73,54,74,94,109,16,31,49,69,50,70,91,106,51,71,92,107,93,108,119,125)(4,10,21,36,22,37,58,78,23,38,59,79,60,80,98,113,20,35,55,75,56,76,95,110,57,77,96,111,97,112,120,126)(15,30,46,66,47,67,88,103,48,68,89,104,90,105,118,124,45,65,85,100,86,101,115,121,87,102,116,122,117,123,127,128);(1,3)(2,9)(4,15)(5,17)(6,18)(7,19)(8,16)(10,30)(11,32)(12,33)(13,34)(14,31)(20,45)(21,46)(22,47)(23,48)(24,52)(25,53)(26,49)(27,54)(28,50)(29,51)(35,65)(36,66)(37,67)(38,68)(39,72)(40,73)(41,69)(42,74)(43,70)(44,71)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,94)(62,91)(63,92)(64,93)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,109)(82,106)(83,107)(84,108)(95,115)(96,116)(97,117)(98,118)(99,119)(110,121)(111,122)(112,123)(113,124)(114,125)(120,127)(126,128);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,20)(9,30)(11,36)(12,37)(13,38)(14,35)(16,45)(17,46)(18,47)(19,48)(24,58)(25,59)(26,55)(27,60)(28,56)(29,57)(31,65)(32,66)(33,67)(34,68)(39,78)(40,79)(41,75)(42,80)(43,76)(44,77)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(61,98)(62,95)(63,96)(64,97)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(81,113)(82,110)(83,111)(84,112)(91,115)(92,116)(93,117)(94,118)(99,120)(106,121)(107,122)(108,123)(109,124)(114,126)(119,127)(125,128);(4,20)(10,35)(15,45)(21,55)(22,56)(23,57)(30,65)(36,75)(37,76)(38,77)(46,85)(47,86)(48,87)(58,95)(59,96)(60,97)(66,100)(67,101)(68,102)(78,110)(79,111)(80,112)(88,115)(89,116)(90,117)(98,120)(103,121)(104,122)(105,123)(113,126)(118,127)(124,128);(3,16)(9,31)(15,45)(17,49)(18,50)(19,51)(30,65)(32,69)(33,70)(34,71)(46,85)(47,86)(48,87)(52,91)(53,92)(54,93)(66,100)(67,101)(68,102)(72,106)(73,107)(74,108)(88,115)(89,116)(90,117)(94,119)(103,121)(104,122)(105,123)(109,125)(118,127)(124,128)Polycyclic presentation:
2^9|a^2=f,f^2=g,g^2=h,h^2=i,e^b=ei,d^c=di272 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532379);