Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129554 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(10,14,15,23)(11,17)(12,20)(16,18,21,24)(19,22);(1,2,4,3)(5,6)(7,8);(1,3,4,2)(6,8)(9,10)(11,18)(12,21)(13,15)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^c=ehi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309664, #72309669, #72309693, … |
| maximal normal subgroup of | #72309664, #72309669, #72309693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55906);