Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134691 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #427 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1808 |
| prime radical | C24 | #107 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
267 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,29)(20,35,37,25)(21,39)(22,42)(23,27)(24,32)(26,30)(28,40)(31,56,58,49)(33,62)(34,51)(36,66,68,69)(38,47,67,65)(41,70)(43,57,63,79)(44,61)(45,72)(46,54,64,76)(48,55)(50,53)(52,60)(59,74,78,77)(71,80,73,75);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,30,53,22,33,41,61,23,34,55,72,40,60,21,32)(18,24,39,52,28,45,48,51,27,44,70,62,42,50,26,29)(20,31,54,75,38,59,68,79,37,58,76,80,67,78,36,57)(25,43,69,74,47,71,64,56,35,63,66,77,65,73,46,49);(3,9)(6,12)(10,15)(13,16)(17,20)(18,25)(19,31)(21,36)(22,38)(23,37)(24,43)(26,46)(27,35)(28,47)(29,49)(30,54)(32,57)(33,59)(34,58)(39,69)(40,67)(41,68)(42,65)(44,63)(45,71)(48,64)(50,73)(51,56)(52,74)(53,75)(55,76)(60,78)(61,79)(62,77)(66,70)(72,80)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41892);