Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461095 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16379 |
| 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 | |
855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,13,8,28)(3,14,29,9,18,32,17,33)(4,10,21,36,22,37,23,42)(5,11,27,40,26,41,25,39)(15,34,51,57,45,56,19,30)(16,46,48,59,49,43,50,31)(20,35,54,60,53,61,24,38)(44,63,47,55,64,58,52,62);(1,3,7,18)(2,9)(4,19,22,51)(5,16,26,49)(6,17,8,29)(10,34)(11,31)(12,14)(13,33)(15,21,45,23)(20,52,53,47)(24,44,54,64)(25,48,27,50)(28,32)(30,36)(35,58)(37,56)(38,62)(39,46)(40,43)(41,59)(42,57)(55,60)(61,63);(1,4,7,22)(2,10,13,37)(3,15,18,45)(5,24,26,54)(6,23,8,21)(9,30,33,57)(11,38,41,60)(12,42,28,36)(14,34,32,56)(16,47,49,52)(17,51,29,19)(20,27,53,25)(31,58,59,63)(35,40,61,39)(43,62,46,55)(44,50,64,48);(1,5)(2,11)(3,16)(4,20)(6,27)(7,26)(8,25)(9,31)(10,35)(12,40)(13,41)(14,43)(15,44)(17,50)(18,49)(19,52)(21,54)(22,53)(23,24)(28,39)(29,48)(30,55)(32,46)(33,59)(34,58)(36,60)(37,61)(38,42)(45,64)(47,51)(56,63)(57,62)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368292);