Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | |
3214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(6,7)(9,10,14,16)(11,20,22,17)(12,18,23,21)(13,24,15,19);(5,6,8,7)(10,16)(11,18)(12,20)(13,15)(17,23)(21,22);(2,4)(5,7,8,6)(9,11,14,22)(10,17,16,20)(12,19,23,24)(13,18,15,21);(9,12)(10,18)(11,19)(13,20)(14,23)(15,17)(16,21)(22,24);(9,13,14,15)(10,19,16,24)(11,21,22,18)(12,20,23,17)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345925);