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 | |
3304 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)(9,10,16,24,15,19,14,18)(11,20,21,17,12,13,22,23);(5,6)(7,8)(9,11)(10,17)(12,15)(13,18)(14,21)(16,22)(19,23)(20,24);(2,4)(6,8)(9,12,15,11)(10,13,19,20)(14,21,16,22)(17,24,23,18);(2,4)(9,13)(10,11)(12,19)(14,23)(15,20)(16,17)(18,21)(22,24);(9,14,15,16)(10,18,19,24)(11,21,12,22)(13,23,20,17)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,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, 7408204);