Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489385 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #121 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2966 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
645 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 645 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 256 |
A minimal faithful permutation representation — 4 generators on 80 points:
(3,9)(4,11)(6,14)(7,12)(10,13)(15,16)(17,18,23,28)(19,29,32,58)(20,25,37,52)(21,38,42,74)(22,43,40,27)(24,45,47,31)(26,53,55,41)(30,59,62,57)(33,64,44,54)(34,65,35,69)(36,72,67,51)(39,56,75,48)(46,60,76,61)(49,71,50,68)(63,78,73,79)(66,77,70,80);(1,2,5,8)(3,6,10,15)(4,12,11,7)(9,16,13,14)(17,19)(18,24)(20,30)(21,39)(22,44)(23,32)(25,46)(26,54)(27,56)(28,47)(29,38)(31,41)(33,40)(34,66)(35,70)(36,73)(37,62)(42,75)(43,48)(45,53)(49,78)(50,79)(51,80)(52,76)(55,64)(57,65)(58,74)(59,69)(60,68)(61,71)(63,67)(72,77);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,11)(12,14)(17,20)(18,25)(19,30)(21,34)(22,36)(23,37)(24,46)(26,49)(27,51)(28,52)(29,57)(31,60)(32,62)(33,63)(35,42)(38,65)(39,66)(40,67)(41,68)(43,72)(44,73)(45,61)(47,76)(48,77)(50,55)(53,71)(54,78)(56,80)(58,59)(64,79)(69,74)(70,75);(17,21,40,53,23,42,22,41)(18,26,43,74,28,55,27,38)(19,31,44,75,32,45,33,39)(20,34,67,71,37,35,36,68)(24,29,56,64,47,58,48,54)(25,49,72,69,52,50,51,65)(30,60,73,70,62,61,63,66)(46,57,80,79,76,59,77,78)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=gi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396681);