Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622825 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6,12,17)(7,18,22,46)(8,14,11,32)(9,15,25,40)(10,28,31,57)(13,34,38,21)(16,24,43,30)(19,44,33,42)(20,45,48,66)(23,39,27,55)(26,52,51,67)(29,35,58,37)(36,59,61,49)(41,50,63,54)(47,65,56,64)(53,60,68,62);(5,7)(6,13)(8,19)(9,20)(10,29)(11,33)(12,22)(14,35)(15,36)(16,42)(17,38)(18,28)(21,30)(23,47)(24,34)(25,48)(26,53)(27,56)(31,58)(32,37)(39,60)(40,61)(41,64)(43,44)(45,52)(46,57)(49,54)(50,59)(51,68)(55,62)(63,65)(66,67);(2,4)(5,8,12,11)(6,14,17,32)(7,19,22,33)(9,23,25,27)(10,24,31,30)(13,35,38,37)(15,39,40,55)(16,28,43,57)(18,44,46,42)(20,47,48,56)(21,29,34,58)(26,50,51,54)(36,60,61,62)(41,52,63,67)(45,65,66,64)(49,53,59,68);(5,9)(6,15)(7,20)(8,23)(10,26)(11,27)(12,25)(13,36)(14,39)(16,41)(17,40)(18,45)(19,47)(21,49)(22,48)(24,50)(28,52)(29,53)(30,54)(31,51)(32,55)(33,56)(34,59)(35,60)(37,62)(38,61)(42,64)(43,63)(44,65)(46,66)(57,67)(58,68);(9,25)(15,40)(20,48)(23,27)(26,51)(36,61)(39,55)(41,63)(45,66)(47,56)(49,59)(50,54)(52,67)(53,68)(60,62)(64,65)Polycyclic presentation:
2^9|a^2=i,c^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,f^b=fh,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530026);