Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 |
| prime radical | C2×#3268 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2694 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)(9,10,14,18)(11,15,13,16)(12,20,22,17)(19,24,21,23);(1,2,3,4)(5,6)(7,8)(9,11,14,13)(10,15,18,16)(12,21,22,19)(17,24,20,23);(6,8)(10,16)(11,13)(12,22)(15,18)(17,23)(20,24);(5,6)(7,8)(9,12,14,22)(10,17,18,20)(11,19,13,21)(15,23,16,24);(6,8)(12,22)(17,20)(19,21)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72342365, #72342369, #72342372 |
| maximal normal subgroup of | #72342365, #72342369, #72342372 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481168);