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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(15,21)(16,24)(18,25)(20,26)(22,27)(23,28);(6,10)(7,11)(13,14)(15,22)(16,20)(17,19)(18,23)(21,27)(24,26)(25,28);(6,10)(9,12)(13,15,16,21,17,23,24,28)(14,18,20,25,19,22,26,27);(1,3,4,2)(14,19)(15,23)(20,26)(21,28)Polycyclic presentation:
2^9|b^2=f,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 | #72256838, #72256852, #72256858, … |
| maximal normal subgroup of | #72256838, #72256852, #72256858, … |
| maximal quotient of | #72264216, #72265066, C2×#7611549 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518745);