Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2847 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14697 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,17,9,19,31,18,32)(4,20,23,34,24,50,22,10)(5,11,25,36,27,37,26,35)(15,29,41,48,43,55,42,38)(16,44,47,56,46,39,45,30)(21,49,51,58,53,57,52,33)(40,60,63,59,62,54,61,64);(1,3,7,17,8,19,6,18)(2,9,12,31,13,32,28,14)(4,15,23,41,24,43,22,42)(5,16,25,47,27,46,26,45)(10,29,20,48,34,55,50,38)(11,30,36,44,37,56,35,39)(21,40,51,63,53,62,52,61)(33,54,49,64,58,60,57,59);(1,4,6,22,8,24,7,23)(2,10,12,20,13,34,28,50)(3,15,18,42,19,43,17,41)(5,21,26,52,27,53,25,51)(9,29,31,48,32,55,14,38)(11,33,36,49,37,58,35,57)(16,40,45,61,46,62,47,63)(30,54,44,64,56,60,39,59);(1,5)(2,11)(3,16)(4,21)(6,26)(7,25)(8,27)(9,30)(10,33)(12,36)(13,37)(14,39)(15,40)(17,47)(18,45)(19,46)(20,49)(22,52)(23,51)(24,53)(28,35)(29,54)(31,44)(32,56)(34,58)(38,59)(41,63)(42,61)(43,62)(48,64)(50,57)(55,60)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,g^b=gi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281077, #72281139, #72281314, … |
| maximal normal subgroup of | #72281077, #72281139, #72281314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300095);