Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130254 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3259 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,21,27,37,26,43,31,23,28,38,25,22,19,29,32)(20,30,33,47,34,24,39,41,35,44,48,46,36,42,45,40);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19)(18,22)(20,30)(21,27)(23,26)(24,36)(25,38)(28,37)(29,31)(32,43)(33,41)(34,42)(35,44)(39,47)(40,48)(45,46);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,20,23,35)(18,24,28,42)(19,30,26,44)(21,33,38,48)(22,36,37,34)(25,40,27,41)(29,45,43,39)(31,46,32,47)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=hi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37658);