Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219498 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6)(7,18)(8,24,28,38)(9,29)(10,16)(11,17)(12,21)(13,27,39,53)(14,31)(15,30)(19,44,47,33)(20,48)(22,35)(23,42)(25,36,46,61)(26,37,56,62)(32,40)(34,51)(41,58)(43,49)(45,65,67,60)(50,63)(52,68,57,55)(54,64,66,59);(1,2,4,3)(5,7,10,21,11,23,30,22)(6,12,15,18,17,35,16,42)(8,25,53,19,28,46,27,47)(9,20,31,49,32,51,58,50)(13,36,24,33,39,61,38,44)(14,34,29,43,41,48,40,63)(26,54,68,45,56,66,55,67)(37,60,52,59,62,65,57,64);(5,8,11,28)(6,13)(7,19)(9,26,32,56)(10,27,30,53)(12,33,35,44)(14,37)(15,38)(16,24)(17,39)(18,36,42,61)(20,45)(21,25)(22,46)(23,47)(29,57)(31,55,58,68)(34,59,48,64)(40,52)(41,62)(43,60,63,65)(49,54)(50,66)(51,67);(1,3,4,2)(5,9,30,58,11,32,10,31)(6,14,16,40,17,41,15,29)(7,20,22,50,23,51,21,49)(8,26,53,68,28,56,27,55)(12,34,42,63,35,48,18,43)(13,37,24,52,39,62,38,57)(19,45,46,66,47,67,25,54)(33,59,61,65,44,64,36,60)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276380);