Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428079 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15377 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1128 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)(3,14)(4,10)(5,23)(6,27)(7,12)(8,13)(9,17)(11,26)(15,35)(16,40)(18,33)(19,49)(20,36)(21,38)(22,39)(24,32)(25,41)(28,42)(29,34)(30,37)(31,45)(43,47)(44,59)(46,58)(48,57)(50,64)(51,62)(52,61)(53,55)(54,60)(56,63);(1,3,6,17,8,18,7,29)(2,9,12,14,13,34,27,33)(4,19,21,15,22,45,30,46)(5,24,25,47,26,48,28,16)(10,35,37,49,39,58,38,31)(11,40,41,57,23,43,42,32)(20,51,52,63,53,44,54,50)(36,56,60,62,55,64,61,59);(1,4)(2,10)(3,15)(5,20)(6,21)(7,30)(8,22)(9,31)(11,36)(12,37)(13,39)(14,35)(16,44)(17,45)(18,46)(19,29)(23,55)(24,50)(25,52)(26,53)(27,38)(28,54)(32,56)(33,58)(34,49)(40,62)(41,60)(42,61)(43,59)(47,51)(48,63)(57,64);(1,5,6,28,8,26,7,25)(2,11,12,42,13,23,27,41)(3,16,17,48,18,47,29,24)(4,20,21,54,22,53,30,52)(9,32,14,43,34,57,33,40)(10,36,37,61,39,55,38,60)(15,44,45,63,46,51,19,50)(31,56,35,59,49,64,58,62)Polycyclic presentation:
2^9|b^2=e,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335275);