Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24267 |
| 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 | |
952 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)(3,15)(4,20)(5,27)(6,32)(7,34)(8,13)(9,14)(10,24)(11,28)(12,16)(17,45)(18,31)(19,33)(21,41)(22,51)(23,36)(25,38)(26,37)(29,48)(30,58)(35,60)(39,42)(40,59)(43,47)(44,57)(46,54)(49,52)(50,55)(53,56)(61,64)(62,63);(1,3,16,14,7,19,33,34,9,12,15,2,8,32,6,13)(4,21,43,37,25,17,45,38,26,47,41,20,24,36,23,10)(5,18,44,40,29,48,59,57,31,27,42,11,30,58,28,39)(22,50,63,61,54,46,64,62,55,51,60,49,53,56,52,35);(1,4)(2,10)(3,17)(5,22)(6,23)(7,25)(8,24)(9,26)(11,35)(12,36)(13,37)(14,38)(15,41)(16,43)(18,46)(19,47)(20,34)(21,32)(27,56)(28,52)(29,54)(30,53)(31,55)(33,45)(39,61)(40,62)(42,60)(44,63)(48,51)(49,57)(50,58)(59,64);(1,5,16,44,7,29,33,59,9,31,15,42,8,30,6,28)(2,11,32,58,13,39,3,18,14,40,19,48,34,57,12,27)(4,22,43,63,25,54,45,64,26,55,41,60,24,53,23,52)(10,35,21,50,37,61,17,46,38,62,47,51,20,49,36,56)Polycyclic presentation:
2^9|b^2=eh,d^2=eh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408056);