Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,15)(12,17)(13,16)(14,18)(19,22)(20,23)(21,24);(1,2,3,4)(5,6)(7,8)(9,11,14,13)(10,15,18,16)(12,20,21,19)(17,23,24,22);(6,8)(10,16)(11,13)(12,21)(15,18)(17,22)(23,24);(5,6)(7,8)(9,12,14,21)(10,17,18,24)(11,19,13,20)(15,22,16,23);(6,8)(12,21)(17,24)(19,20)(22,23)Polycyclic presentation:
2^9|b^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72342365, #72342370, #72342372 |
| maximal normal subgroup of | #72342365, #72342370, #72342372 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481403);