Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
3278 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)(9,10,15,23,14,18,13,19)(11,12,22,16,20,17,21,24);(5,6)(7,8)(9,11)(10,16)(12,23)(13,22)(14,20)(15,21)(17,19)(18,24);(2,4)(6,8)(9,11)(10,17)(12,18)(13,21)(14,20)(15,22)(16,19)(23,24);(2,4)(9,12,14,17)(10,11,18,20)(13,16,15,24)(19,22,23,21);(10,18)(11,20)(19,23)(21,22)Polycyclic presentation:
2^9|a^2=fi,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408056);