Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131804 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #61502 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
568 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 72 points:
(2,5,6,7)(9,10,15,22,17,23,16,38)(11,24,28,42,29,55,39,51)(12,30,33,47,34,67,40,66)(13,21,36,52,37,53,14,20)(18,35,43,63,45,27,44,26)(19,46,48,68,50,31,49,32)(25,56,59,41,61,70,60,71)(54,64,72,62,69,57,65,58);(1,2)(3,7)(4,6)(5,8)(9,11)(10,18)(12,25)(13,35)(14,26)(15,28)(16,39)(17,29)(19,41)(20,51)(21,24)(22,43)(23,45)(27,37)(30,65)(31,62)(32,57)(33,59)(34,61)(36,63)(38,44)(40,60)(42,52)(46,58)(47,54)(48,70)(49,56)(50,71)(53,55)(64,68)(66,69)(67,72);(9,12,17,34)(10,19,23,50)(11,25,29,61)(13,32,37,68)(14,31,36,46)(15,33,16,40)(18,41,45,71)(20,30,52,67)(21,47,53,66)(22,48,38,49)(24,54,55,69)(26,62,63,58)(27,64,35,57)(28,59,39,60)(42,72,51,65)(43,70,44,56)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,e^2=g,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,h^a=hi,d^b=dh,e^b=eg,f^b=fghi,g^b=gi,h^b=hi,d^c=dg,e^c=egi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39003);