Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574270 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,14,18)(11,19,17,13)(12,16,22,15)(20,23,21,24);(5,6)(7,8)(9,11,13,10,14,17,19,18)(12,21,23,16,22,20,24,15);(1,2,3,4)(6,8)(9,12)(10,15)(11,20)(13,23)(14,22)(16,18)(17,21)(19,24);(5,6)(7,8)(9,12)(10,16)(11,21)(13,23)(14,22)(15,18)(17,20)(19,24);(6,8)(12,22)(15,16)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490354);