Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(15,20)(16,26)(19,23)(25,27);(2,3)(6,9)(10,11)(13,14,17,21)(15,23,20,19)(16,27,26,25)(18,24,28,22);(1,2,4,3)(13,15)(14,19)(16,22)(17,20)(18,25)(21,23)(24,26)(27,28);(1,3,4,2)(13,16,18,23,17,26,28,19)(14,20,22,27,21,15,24,25);(5,7,8,12)(6,10,9,11)(15,20)(16,26)(19,23)(25,27)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72280333, #72280484, #72282712, … |
| maximal normal subgroup of | #72280333, #72280484, #72282712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521043);