Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455697 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16268 |
| 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 | |
1068 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)(3,13)(4,9)(5,10)(6,25)(7,12)(8,16)(11,28)(14,40)(15,45)(17,42)(18,52)(19,32)(20,38)(21,34)(22,56)(23,39)(24,37)(26,33)(27,36)(29,49)(30,47)(31,51)(35,55)(41,48)(43,64)(44,59)(46,63)(50,61)(53,60)(54,62)(57,58);(1,3,6,16,7,17,28,51)(2,8,11,13,12,31,25,42)(4,18,20,49,21,14,26,44)(5,15,27,50,24,48,23,47)(9,29,33,52,34,59,38,40)(10,30,39,41,37,61,36,45)(19,53,55,46,54,43,22,57)(32,58,56,64,62,63,35,60);(1,4,7,21)(2,9,12,34)(3,14,17,18)(5,22,24,55)(6,26,28,20)(8,29,31,59)(10,35,37,56)(11,38,25,33)(13,40,42,52)(15,46,48,57)(16,49,51,44)(19,27,54,23)(30,60,61,64)(32,39,62,36)(41,63,45,58)(43,50,53,47);(1,5)(2,10)(3,15)(4,19)(6,27)(7,24)(8,30)(9,32)(11,39)(12,37)(13,41)(14,43)(16,50)(17,48)(18,53)(20,55)(21,54)(22,26)(23,28)(25,36)(29,58)(31,61)(33,56)(34,62)(35,38)(40,60)(42,45)(44,57)(46,49)(47,51)(52,64)(59,63)Polycyclic presentation:
2^9|b^2=ef,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362894);