Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | #34736 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,13,16,14,17,12,21)(11,15,20,23,18,22,19,24);(2,5)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24);(11,18)(15,22)(19,20)(23,24);(2,5)(6,8)(9,12,14,13)(10,16,17,21)(11,19,18,20)(15,23,22,24)Polycyclic presentation:
2^9|a^2=f,d^2=hi,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,f^b=fg,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285284, #72285386, #72307917, … |
| maximal normal subgroup of | #72285284, #72285386, #72307917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387985);