Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(9,10,15,12,14,18,13,16)(11,19,23,24,20,17,22,21);(1,2)(3,4)(10,16,18,12)(11,20)(17,21,19,24)(22,23);(6,8)(9,11,13,22,14,20,15,23)(10,17,16,24,18,19,12,21);(2,4)(9,12,13,10,14,16,15,18)(11,21,22,17,20,24,23,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=fg,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^b=dei,e^d=ei,h^d=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72310066, D4×#1765, #72310141 |
| maximal normal subgroup of | #72310066, D4×#1765, #72310141 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264186);