Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21851 |
| prime radical | #3444 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10,5,8,16,22,6,9,17,23,18,21,29,31)(3,11,13,24,14,25,27,32,12,7,15,20,26,19,28,30);(1,3)(2,7)(4,15)(5,14)(6,12)(8,19)(9,11)(10,24)(13,17)(16,28)(18,26)(20,23)(21,25)(22,32)(27,29)(30,31);(7,11)(19,25)(20,24)(30,32);(2,8,9,21)(7,19,11,25)(10,22,23,31)(20,30,24,32)Polycyclic presentation:
2^9|a^2=g,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=ce,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72217272, #72217274, #72224957, … |
| maximal normal subgroup of | #72217272, #72217274, #72224957, … |
| maximal quotient of | C2×#496469 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403665);