Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500749 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24248 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,16)(4,23)(5,28)(6,10)(7,32)(8,15)(9,19)(11,24)(12,29)(13,30)(14,31)(17,36)(18,50)(20,47)(21,39)(22,38)(25,41)(26,40)(27,56)(33,44)(34,45)(35,55)(37,49)(42,57)(43,58)(46,52)(48,61)(51,64)(53,60)(54,62)(59,63);(1,3,5,19,8,22,29,21)(2,9,12,16,15,39,28,38)(4,18,24,52,27,55,58,54)(6,20,30,17,31,49,7,33)(10,36,32,47,14,44,13,37)(11,35,23,46,43,50,56,62)(25,51,26,48,57,53,34,59)(40,60,41,61,45,64,42,63);(2,10)(3,17)(4,24)(5,29)(6,30)(7,31)(9,16)(11,41)(12,13)(14,15)(18,51)(19,20)(21,49)(22,33)(23,40)(25,57)(27,58)(28,32)(37,47)(38,39)(42,43)(45,56)(46,62)(48,54)(52,59)(53,55)(60,63)(61,64);(1,4,5,24,8,27,29,58)(2,11,12,23,15,43,28,56)(3,18,19,52,22,55,21,54)(6,25,30,26,31,57,7,34)(9,35,16,46,39,50,38,62)(10,40,32,41,14,45,13,42)(17,48,49,53,33,59,20,51)(36,60,47,61,44,64,37,63)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=ghi,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407945);