Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #472803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16837 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,15,18,9)(4,20,24,41)(5,25,27,40)(6,14,8,12)(10,30,37,23)(11,28,39,29)(16,44,21,52)(17,48,50,59)(19,34,31,42)(22,38,26,36)(32,46,35,47)(33,49,43,57)(45,60,53,58)(51,56,55,54)(61,62,64,63);(1,3,6,19,7,18,8,31)(2,9,12,34,13,15,14,42)(4,21,30,46,24,16,23,47)(5,17,28,49,27,50,29,57)(10,35,41,52,37,32,20,44)(11,33,40,59,39,43,25,48)(22,53,56,62,26,45,54,63)(36,60,55,64,38,58,51,61);(1,4)(2,10)(3,16)(5,26)(6,30)(7,24)(8,23)(9,32)(11,38)(12,41)(13,37)(14,20)(15,35)(17,45)(18,21)(19,47)(22,27)(25,55)(28,54)(29,56)(31,46)(33,58)(34,44)(36,39)(40,51)(42,52)(43,60)(48,64)(49,63)(50,53)(57,62)(59,61);(1,5,6,28,7,27,8,29)(2,11,12,40,13,39,14,25)(3,17,19,49,18,50,31,57)(4,22,23,54,24,26,30,56)(9,33,34,59,15,43,42,48)(10,36,20,51,37,38,41,55)(16,45,46,62,21,53,47,63)(32,58,52,64,35,60,44,61)Polycyclic presentation:
2^9|a^2=i,b^2=egh,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380000);