Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423178 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15312 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,17,38)(4,18,21,9)(5,10,24,37)(6,25,29,11)(8,26,32,16)(14,41,45,58)(15,42,46,35)(19,33,53,50)(20,34,27,39)(22,55,48,31)(23,40,28,36)(30,44,49,47)(43,62,63,59)(51,64,56,57)(52,61,54,60);(1,3)(2,8)(4,14)(5,22)(6,26)(7,17)(9,30)(10,35)(11,38)(12,32)(13,25)(15,23)(16,29)(18,49)(19,51)(20,47)(21,45)(24,48)(27,44)(28,46)(31,36)(33,59)(34,41)(37,42)(39,58)(40,55)(43,52)(50,62)(53,56)(54,63)(57,60)(61,64);(1,4,6,27,7,21,29,20)(2,9,11,39,12,18,25,34)(3,14,26,44,17,45,16,47)(5,19,28,52,24,53,23,54)(8,30,38,58,32,49,13,41)(10,33,40,60,37,50,36,61)(15,43,22,56,46,63,48,51)(31,57,35,62,55,64,42,59);(1,5)(2,10)(3,15)(4,19)(6,28)(7,24)(8,31)(9,33)(11,40)(12,37)(13,42)(14,43)(16,48)(17,46)(18,50)(20,54)(21,53)(22,26)(23,29)(25,36)(27,52)(30,57)(32,55)(34,61)(35,38)(39,60)(41,59)(44,56)(45,63)(47,51)(49,64)(58,62)Polycyclic presentation:
2^9|a^2=i,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330375);