Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8316 |
| prime radical | #3529 | C4 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,6,12,23,35,7,11,24,34,25,36,50,59)(3,13,16,38,17,39,45,61,14,37,41,54,42,60,62,64)(4,18,20,33,21,46,48,58,19,9,22,30,47,31,49,56)(8,26,28,51,29,52,55,15,27,43,32,44,53,63,57,40);(1,3)(2,8)(4,15)(5,16)(6,17)(7,14)(9,13)(10,32)(11,27)(12,29)(18,37)(19,40)(20,43)(21,44)(22,26)(23,45)(24,41)(25,42)(28,34)(30,54)(31,39)(33,38)(35,57)(36,53)(46,60)(47,51)(48,63)(49,52)(50,62)(55,59)(56,64)(58,61);(3,14)(8,27)(9,18)(15,40)(16,41)(17,42)(26,43)(28,32)(29,53)(30,33)(31,46)(44,51)(45,62)(52,63)(55,57)(56,58)Polycyclic presentation:
2^9|a^2=f,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,e^b=egi,f^b=fg,d^c=dg,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239956, #72250025, #72250039, … |
| maximal normal subgroup of | #72239956, #72250025, #72250039, … |
| maximal quotient of | #72239959, #72280076 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36810);