Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1744 |
| prime radical | C23 | #431 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
252 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16)(17,18)(19,27,30,40)(20,24)(21,25)(22,26)(23,29,35,43)(28,39,42,47)(31,36)(32,37)(33,38)(34,41,45,48)(44,46);(17,19)(18,23,26,35)(20,28)(21,29)(22,30)(24,34,37,45)(25,27,38,40)(31,41)(32,42)(33,43)(36,39,46,47)(44,48);(2,6)(7,12)(8,13)(14,16)(17,20,21,31,22,32,33,44)(18,24,25,36,26,37,38,46)(19,28,29,41,30,42,43,48)(23,34,27,39,35,45,40,47)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=d,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fg,e^b=eg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72210248, #72210250, C2×#128226 |
| maximal normal subgroup of | #72210248, #72210250, C2×#128226 |
| maximal quotient of | #72210241, C2×#128226, #72210594 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35422);