Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480250 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10,13,17,14,15,21,22)(11,18,19,24,12,16,20,23);(10,15)(11,12)(19,20)(23,24);(2,6)(7,8)(9,11,14,12)(10,16,15,18)(13,19,21,20)(17,23,22,24);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,16,15,18)(13,20,21,19)(17,23,22,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,d^b=di,f^b=fg,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282431, #72282466, #72282486, … |
| maximal normal subgroup of | #72282431, #72282466, #72282486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387582);