Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3027 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)(5,6,11,14)(7,18,22,46)(8,24,28,15)(9,29,32,45)(10,33,12,17)(13,23,36,21)(16,34,39,31)(19,40,50,47)(20,38,53,63)(25,56,55,37)(26,58,30,61)(27,42,35,43)(41,59,44,60)(48,54,66,52)(49,65,51,64)(57,68,62,67);(1,2)(3,4)(5,7,11,22)(6,13,14,36)(8,25,28,55)(9,20,32,53)(10,23,12,21)(15,40,24,47)(16,38,39,63)(17,46,33,18)(19,35,50,27)(26,57,30,62)(29,48,45,66)(31,54,34,52)(37,42,56,43)(41,67,44,68)(49,59,51,60)(58,65,61,64);(2,4)(6,14)(13,36)(15,24)(16,39)(17,33)(18,46)(29,45)(37,56)(38,63)(40,47)(41,44)(42,43)(48,66)(58,61)(64,65)(67,68);(2,4)(5,8)(6,15)(7,19)(9,30)(10,27)(11,28)(12,35)(13,37)(14,24)(16,44)(17,42)(18,47)(20,51)(21,25)(22,50)(23,55)(26,32)(29,58)(31,60)(33,43)(34,59)(36,56)(38,65)(39,41)(40,46)(45,61)(48,67)(49,53)(52,57)(54,62)(63,64)(66,68);(5,9)(6,16)(7,20)(8,26)(10,34)(11,32)(12,31)(13,38)(14,39)(15,41)(17,45)(18,48)(19,49)(21,54)(22,53)(23,52)(24,44)(25,57)(27,60)(28,30)(29,33)(35,59)(36,63)(37,64)(40,67)(42,58)(43,61)(46,66)(47,68)(50,51)(55,62)(56,65)Polycyclic presentation:
2^9|a^2=i,b^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,h^b=hi,e^d=ei,h^d=hi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421382);