Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,8)(9,10)(11,13)(12,15)(14,16)(18,20)(19,23);(1,2,4,3)(10,11)(14,15)(17,18,21,20)(19,24,23,22);(9,10,13,11)(12,14,16,15)(17,18,21,20)(19,22,23,24);(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19,21,23)(18,22,20,24);(1,3,4,2)(9,12)(10,14)(11,15)(13,16)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^9|b^2=fh,c^2=g,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232325, #72232337, #72232420, … |
| maximal normal subgroup of | #72232325, #72232337, #72232420, … |
| maximal quotient of | #72298511, #72330736, #72332724 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 757343);