Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C32 | C24 |
| commutator, socle | C2 | C23×C32 |
| prime radical | C2×#41 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,4,11,5,12,20,39,6,13,21,40,24,42,55,81,7,14,22,41,25,43,56,82,27,44,58,83,61,84,95,114,8,10,23,35,26,36,57,75,28,37,59,76,62,78,96,110,29,38,60,77,63,79,97,111,64,80,98,112,99,113,120,126)(3,9,16,31,17,32,49,69,18,33,50,70,52,72,91,106,19,34,51,71,53,73,92,107,54,74,93,108,94,109,119,125,15,30,45,65,46,66,85,100,47,67,86,101,88,103,115,121,48,68,87,102,89,104,116,122,90,105,117,123,118,124,127,128);(1,3)(2,9)(4,16)(5,17)(6,18)(7,19)(8,15)(10,30)(11,31)(12,32)(13,33)(14,34)(20,49)(21,50)(22,51)(23,45)(24,52)(25,53)(26,46)(27,54)(28,47)(29,48)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(55,91)(56,92)(57,85)(58,93)(59,86)(60,87)(61,94)(62,88)(63,89)(64,90)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(95,119)(96,115)(97,116)(98,117)(99,118)(110,121)(111,122)(112,123)(113,124)(114,125)(120,127)(126,128);(3,15)(9,30)(16,45)(17,46)(18,47)(19,48)(31,65)(32,66)(33,67)(34,68)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(91,115)(92,116)(93,117)(94,118)(106,121)(107,122)(108,123)(109,124)(119,127)(125,128);(2,10)(9,30)(11,35)(12,36)(13,37)(14,38)(31,65)(32,66)(33,67)(34,68)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(81,110)(82,111)(83,112)(84,113)(106,121)(107,122)(108,123)(109,124)(114,126)(125,128)Polycyclic presentation:
2^9|a^2=e,e^2=f,f^2=g,g^2=h,h^2=i,d^a=di,c^b=ci272 conjugacy classes — view the full character table.
| maximal subgroup of | #72321329, #72321377, #72321383, … |
| maximal normal subgroup of | #72321329, #72321377, #72321383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420504);