Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #13742 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D32 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C2 | C4×D32 |
| prime radical | #13741 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C8 ◁ C4×C16 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,6,14,10,17,9,44)(3,18,21,66,24,68,45,67)(4,25,27,16,8,12,29,53)(5,30,33,89,37,91,36,100)(7,39,38,54,42,104,41,81)(11,46,48,73,51,108,50,20)(13,55,57,98,59,97,107,34)(15,61,60,28,63,87,103,32)(19,69,23,77,43,65,78,124)(22,75,74,126,76,125,105,120)(26,82,84,90,35,88,86,127)(31,62,92,121,95,123,101,122)(40,56,102,117,106,119,71,118)(47,109,110,70,79,72,64,93)(49,99,111,128,113,83,112,94)(52,114,115,85,80,116,58,96);(1,3)(2,11)(4,19)(5,31)(6,21)(7,40)(8,43)(9,45)(10,24)(12,47)(13,56)(14,48)(15,62)(16,64)(17,51)(18,30)(20,34)(22,32)(23,27)(25,79)(26,83)(28,76)(29,78)(33,92)(35,99)(36,101)(37,95)(38,102)(39,49)(41,71)(42,106)(44,50)(46,55)(52,75)(53,110)(54,111)(57,117)(58,120)(59,119)(60,121)(61,74)(63,123)(65,88)(66,89)(67,100)(68,91)(69,82)(70,85)(72,116)(73,98)(77,90)(80,125)(81,112)(84,94)(86,128)(87,105)(93,96)(97,108)(103,122)(104,113)(107,118)(109,114)(115,126)(124,127);(1,4,6,27,10,8,9,29)(2,12,14,53,17,25,44,16)(3,19,21,23,24,43,45,78)(5,26,33,84,37,35,36,86)(7,28,38,87,42,32,41,61)(11,47,48,110,51,79,50,64)(13,52,57,115,59,80,107,58)(15,54,60,104,63,81,103,39)(18,65,66,124,68,69,67,77)(20,70,46,72,73,93,108,109)(22,71,74,40,76,102,105,106)(30,88,89,127,91,82,100,90)(31,83,92,94,95,99,101,128)(34,85,55,116,98,96,97,114)(49,62,111,121,113,123,112,122)(56,75,117,126,119,125,118,120)Polycyclic presentation:
2^9|a^2=e,c^2=e,d^2=fg,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=df,f^b=fg,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72230739, #72230770, #72230775 |
| maximal normal subgroup of | #72230739, #72230770, #72230775 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60792);