Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,7)(10,15)(12,19)(13,20)(16,22)(17,23)(21,24);(2,4)(6,7)(9,10,13,15,14,16,20,22)(11,12,17,19,18,21,23,24);(5,6,8,7)(9,11,14,18)(10,12,16,21)(13,17,20,23)(15,19,22,24);(1,2,3,4)(5,7,8,6)(10,16)(11,18)(15,22)(17,23);(9,12,20,24,14,21,13,19)(10,17,22,11,16,23,15,18)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=hi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521232);