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 | |
3024 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)(7,18)(8,24)(9,29)(10,32)(11,14)(12,17)(13,21)(15,28)(16,30)(19,39)(20,37)(22,44)(23,35)(25,54)(26,56)(27,41)(31,43)(33,38)(34,42)(36,52)(40,59)(45,48)(46,49)(47,66)(50,61)(51,63)(53,57)(55,65)(58,60)(62,68)(64,67);(1,2)(3,4)(5,7,11,22)(6,13,14,35)(8,25,28,52)(9,20,31,50)(10,23,12,21)(15,39,24,45)(16,37,38,61)(17,44,32,18)(19,34,48,27)(26,55,58,67)(29,46,43,63)(30,51,33,49)(36,41,54,42)(40,64,53,65)(47,57,68,59)(56,66,60,62);(2,4)(6,14)(13,35)(15,24)(16,38)(17,32)(18,44)(29,43)(36,54)(37,61)(39,45)(40,53)(41,42)(46,63)(56,60)(62,66)(64,65);(2,4)(5,8,11,28)(6,15,14,24)(7,19,22,48)(9,26,31,58)(10,27,12,34)(13,36,35,54)(16,40,38,53)(17,41,32,42)(18,45,44,39)(20,47,50,68)(21,25,23,52)(29,60,43,56)(30,57,33,59)(37,62,61,66)(46,65,63,64)(49,67,51,55);(5,9)(6,16)(7,20)(8,26)(10,33)(11,31)(12,30)(13,37)(14,38)(15,40)(17,43)(18,46)(19,47)(21,51)(22,50)(23,49)(24,53)(25,55)(27,59)(28,58)(29,32)(34,57)(35,61)(36,62)(39,64)(41,56)(42,60)(44,63)(45,65)(48,68)(52,67)(54,66)Polycyclic presentation:
2^9|b^2=i,d^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,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, 7421376);