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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#97 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
436 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 52 points:
(1,2)(3,4)(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)(21,22)(23,33,35,44)(24,30)(25,39)(26,32)(27,29)(28,34,42,37)(31,41)(36,50,49,47)(38,46)(40,45)(43,51,52,48);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,36,26,37,40,51,27,35,41,49,24,34,38,48)(22,28,31,43,32,44,46,50,29,42,39,52,30,33,45,47);(7,13)(10,16)(14,19)(17,20)(22,29)(28,42)(30,32)(31,39)(33,44)(43,52)(45,46)(47,50)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fi,c^b=cg,e^b=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33771);