Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2668 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,15)(4,21)(5,27)(6,12)(7,13)(8,14)(9,19)(10,25)(11,31)(16,32)(17,40)(18,35)(20,34)(22,51)(23,50)(24,39)(26,38)(28,33)(29,42)(30,41)(36,45)(37,54)(43,47)(44,58)(46,49)(48,59)(52,62)(53,56)(55,61)(57,60)(63,64);(1,3)(2,9)(4,22)(5,28)(6,18)(7,19)(8,20)(10,36)(11,40)(12,15)(13,34)(14,35)(16,24)(17,30)(21,49)(23,44)(25,46)(26,45)(27,43)(29,47)(31,48)(32,38)(33,41)(37,58)(39,51)(42,59)(50,60)(52,54)(53,57)(55,63)(56,64)(61,62);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32)(11,37)(12,21)(13,38)(14,39)(15,36)(17,44)(18,45)(19,22)(20,46)(27,56)(28,57)(29,53)(30,54)(31,55)(33,58)(34,51)(35,49)(40,62)(41,50)(42,61)(43,60)(47,63)(48,52)(59,64);(1,5)(2,11)(3,17)(4,23)(6,29)(7,30)(8,31)(9,33)(10,37)(12,41)(13,42)(14,27)(15,43)(16,44)(18,28)(19,48)(20,47)(21,50)(22,52)(24,53)(25,54)(26,55)(32,58)(34,40)(35,59)(36,60)(38,61)(39,56)(45,57)(46,63)(49,64)(51,62);(1,6,8,7)(2,12,14,13)(3,18,20,19)(4,24,26,25)(5,29,31,30)(9,15,35,34)(10,21,39,38)(11,41,27,42)(16,45,46,22)(17,28,47,48)(23,53,55,54)(32,36,49,51)(33,43,59,40)(37,50,56,61)(44,57,63,52)(58,60,64,62)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407958);