Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220210 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,20,15,19,28,43)(11,16,24,41,25,39,23,40)(12,26,31,45,32,17,30,46)(13,33,35,47,36,18,34,48)(21,49,27,57,51,37,55,42)(22,52,53,65,54,38,56,64)(29,44,60,67,61,58,62,68)(50,63,70,72,71,69,59,66);(1,3,5,2)(4,7,8,6)(9,11,15,25)(10,16,19,39)(12,27,32,55)(13,22,36,54)(14,24,28,23)(17,42,26,57)(18,38,33,52)(20,41,43,40)(21,31,51,30)(29,59,61,70)(34,56,35,53)(37,46,49,45)(44,66,58,72)(47,65,48,64)(50,62,71,60)(63,68,69,67);(1,4,5,8)(2,6,3,7)(9,12,28,31,15,32,14,30)(10,17,43,46,19,26,20,45)(11,21,24,55,25,51,23,27)(13,29,35,62,36,61,34,60)(16,37,41,57,39,49,40,42)(18,44,48,68,33,58,47,67)(22,50,56,70,54,71,53,59)(38,63,65,72,52,69,64,66);(2,3)(6,7)(9,13)(10,18)(11,22)(12,29)(14,34)(15,36)(16,38)(17,44)(19,33)(20,47)(21,50)(23,53)(24,56)(25,54)(26,58)(27,59)(28,35)(30,60)(31,62)(32,61)(37,63)(39,52)(40,64)(41,65)(42,66)(43,48)(45,67)(46,68)(49,69)(51,71)(55,70)(57,72)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=egi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293165);