Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C24 |
| commutator, socle | C23 | C23×C8 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,20,17,21,24,27,18,15,19,22,25,23,26,28);(1,2,3,4)(13,15,16,22,17,23,24,28,18,14,19,20,25,21,26,27);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(14,15)(20,22)(21,23)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=di152 conjugacy classes — view the full character table.
| maximal subgroup of | #72217246, #72224980, #72225017, … |
| maximal normal subgroup of | #72217246, #72224980, #72225017, … |
| maximal quotient of | #72217181, #72217183, #72287763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402687);