Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,14,17)(11,19,21,16)(12,22,20,15)(13,24,23,18);(5,6)(7,8)(9,11,14,21)(10,15,17,22)(12,23,20,13)(16,24,19,18);(2,4)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,20)(21,22)(23,24);(2,4)(9,12,13,11,14,20,23,21)(10,16,18,15,17,19,24,22);(10,17)(15,22)(16,19)(18,24)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,c^b=ch,d^b=df,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409320);