Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232253 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,23,20,22);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(19,20)(22,23);(1,4,5,3)(2,6,7,8)(13,16)(18,22)(19,20)(23,24);(17,19,21,20)(18,23,24,22)Polycyclic presentation:
2^9|a^2=eh,b^2=e,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,d^c=dh,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309181, #72309187, #72309232, … |
| maximal normal subgroup of | #72309181, #72309187, #72309232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 148404);