Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500740 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24236 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1020 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)(3,18)(4,24)(5,15)(6,16)(7,31)(8,34)(9,14)(10,17)(11,21)(12,28)(13,30)(19,53)(20,41)(22,52)(23,49)(25,59)(26,45)(27,46)(29,48)(32,39)(33,50)(35,47)(36,42)(37,44)(38,61)(40,58)(43,57)(51,54)(55,63)(56,62)(60,64);(1,3,6,21,10,23,9,36)(2,11,14,18,17,42,16,49)(4,25,27,54,29,61,37,60)(5,20,30,58,7,32,8,22)(12,43,45,63,47,53,50,62)(13,39,15,40,34,41,31,52)(19,28,55,33,57,35,56,26)(24,51,44,59,48,64,46,38);(1,4)(2,12)(3,19)(5,26)(6,27)(7,33)(8,35)(9,37)(10,29)(11,38)(13,44)(14,45)(15,48)(16,50)(17,47)(18,51)(20,54)(21,55)(22,25)(23,57)(24,31)(28,30)(32,60)(34,46)(36,56)(39,62)(40,43)(41,63)(42,59)(49,64)(52,53)(58,61);(1,5,6,30,10,7,9,8)(2,13,14,15,17,34,16,31)(3,20,21,58,23,32,36,22)(4,26,27,28,29,33,37,35)(11,39,18,40,42,41,49,52)(12,44,45,48,47,46,50,24)(19,54,55,61,57,60,56,25)(38,62,51,43,59,63,64,53)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=fhi,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407936);