Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2688 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1892 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)(3,14)(4,19)(5,23)(6,12)(7,27)(8,13)(9,18)(10,21)(11,25)(15,42)(16,47)(17,34)(20,36)(22,37)(24,56)(26,40)(28,33)(29,38)(30,41)(31,44)(32,49)(35,46)(39,48)(43,57)(45,50)(51,54)(52,58)(53,59)(55,60)(61,63)(62,64);(1,3,7,28,8,18,6,17)(2,9,12,33,13,14,27,34)(4,15,21,44,22,45,29,46)(5,24,25,49,26,16,30,48)(10,35,19,50,38,31,37,42)(11,39,23,47,41,32,40,56)(20,52,54,62,55,43,53,63)(36,57,59,64,60,58,51,61);(1,4)(2,10)(3,15,18,45)(5,20)(6,21)(7,29)(8,22)(9,31,14,35)(11,36)(12,37)(13,38)(16,43,24,52)(17,44,28,46)(19,27)(23,51)(25,53)(26,55)(30,54)(32,57,39,58)(33,50,34,42)(40,59)(41,60)(47,64,56,61)(48,62,49,63);(1,5)(2,11)(3,16)(4,20)(6,25)(7,30)(8,26)(9,32)(10,36)(12,40)(13,41)(14,39)(15,43)(17,48)(18,24)(19,51)(21,53)(22,55)(23,27)(28,49)(29,54)(31,57)(33,47)(34,56)(35,58)(37,59)(38,60)(42,61)(44,62)(45,52)(46,63)(50,64);(1,6,8,7)(2,12,13,27)(3,17,18,28)(4,21,22,29)(5,25,26,30)(9,33,14,34)(10,37,38,19)(11,40,41,23)(15,44,45,46)(16,48,24,49)(20,53,55,54)(31,50,35,42)(32,47,39,56)(36,59,60,51)(43,62,52,63)(57,64,58,61)Polycyclic presentation:
2^9|b^2=fg,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,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,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411871);