Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,8,6,7,5,3)(9,10)(11,15,16,13)(12,14);(1,3,5,7,6,8,4,2)(10,13)(14,15);(2,7)(3,8)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=h,f^2=h,b^a=bd,c^a=ch,d^a=dg,g^a=gi,c^b=ch,e^b=eg,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72279561, #72279953, #72279970 |
| maximal normal subgroup of | #72279561, #72279953, #72279970 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34331);