Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10)(11,18)(12,16)(13,15)(14,20)(17,21)(19,24)(22,23);(1,2,4,3)(5,6)(7,8)(9,10,12,16)(11,18,19,24)(13,15,21,17)(14,22,23,20);(9,11,12,19)(10,14,16,23)(13,22,21,20)(15,18,17,24);(6,8)(10,15)(13,21)(14,18)(16,17)(20,22)(23,24);(1,3,4,2)(10,16)(14,23)(15,17)(18,24)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426300);