Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1404 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)(3,7)(4,6)(5,8)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24);(1,3)(2,5,6,7)(4,8);(9,11,12,17,13,18,19,23)(10,14,16,21,15,20,22,24);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,f^2=i,b^a=be,d^a=di,g^a=gh,e^b=eh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72279972, #72283943, #72310544, … |
| maximal normal subgroup of | #72279972, #72283943, #72310544, … |
| maximal quotient of | C2×#182072 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89268);