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 | #1807 |
| prime radical | C24 | #106 |
| 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)(19,29,32,50)(20,34,36,25)(21,38)(22,41)(23,27)(24,31,44,56)(26,39)(28,42)(30,52)(33,59,58,51)(35,63,64,67)(37,47,66,62)(40,68)(43,53)(45,72,70,57)(46,65,61,79)(48,69)(49,55)(54,73)(60,77)(71,80)(74,76)(75,78);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,31,22,33,39,57,23,32,40,56,42,58,69,72)(18,24,26,29,28,45,38,51,27,44,48,50,41,70,68,59)(20,30,35,53,37,54,65,76,36,55,64,77,66,78,79,80)(25,43,46,49,47,71,67,73,34,60,61,52,62,74,63,75);(3,9)(6,12)(10,15)(13,16)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,43)(26,46)(27,34)(28,47)(29,49)(31,53)(32,55)(33,54)(38,67)(39,65)(40,64)(41,62)(42,66)(44,60)(45,71)(48,61)(50,52)(51,73)(56,77)(57,76)(58,78)(59,75)(63,68)(69,79)(70,74)(72,80)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,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^b=fi,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41887);