Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7)(4,6)(5,8)(9,10,11,14,12,13,15,16);(2,6,8,7)(10,13);(1,3,5,4)(2,7,8,6)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gh,d^2=gh,e^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,e^b=eh,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231150, #72231177, #72231210, … |
| maximal normal subgroup of | #72231150, #72231177, #72231210, … |
| maximal quotient of | C2×#127207, #72261314, #72262212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34403);