Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #490474 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| 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 | |
396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 80 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,28)(19,29,34,54)(20,25,35,46)(21,38,40,66)(22,41,39,27)(24,42,45,32)(26,49,50,30)(31,51,57,73)(33,61,60,53)(36,64,62,79)(37,65,63,48)(43,67,69,58)(44,55,52,68)(47,72,71,56)(59,78,77,75)(70,76,74,80);(17,19,30,55,22,33,40,42,23,34,49,68,39,60,21,32)(18,24,38,53,27,44,50,54,28,45,66,61,41,52,26,29)(20,31,56,76,37,59,62,67,35,57,72,80,63,77,36,58)(25,43,64,75,48,70,71,73,46,69,79,78,65,74,47,51);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,25)(19,31)(21,36)(22,37)(23,35)(24,43)(26,47)(27,48)(28,46)(29,51)(30,56)(32,58)(33,59)(34,57)(38,64)(39,63)(40,62)(41,65)(42,67)(44,70)(45,69)(49,72)(50,71)(52,74)(53,75)(54,73)(55,76)(60,77)(61,78)(66,79)(68,80);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(20,35)(25,46)(31,57)(36,62)(37,63)(43,69)(47,71)(48,65)(51,73)(56,72)(58,67)(59,77)(64,79)(70,74)(75,78)(76,80)Polycyclic presentation:
2^9|a^2=i,b^2=eg,d^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^c=di,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397848);