Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1218 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)(3,6)(4,8)(5,7)(9,10)(11,21,25,16)(12,27)(13,32,33,18)(14,19)(15,20)(17,30)(22,36)(23,37,51,45)(24,38)(26,47,48,39)(28,40,54,55)(29,41,58,56)(31,44)(34,60,49,43)(35,42)(46,52)(50,67)(53,63)(57,72)(59,65,61,66)(62,70)(64,71)(68,69);(1,3,5,4)(2,6,7,8)(9,11,22,48,14,25,15,26)(10,16,36,47,19,21,20,39)(12,28,35,23,30,54,31,51)(13,24,49,69,33,52,34,53)(17,40,44,37,27,55,42,45)(18,38,60,68,32,46,43,63)(29,57,61,50,58,71,59,70)(41,64,66,62,56,72,65,67);(2,7)(6,8)(9,12,14,30)(10,17,19,27)(11,23,25,51)(13,29,33,58)(15,35,22,31)(16,37,21,45)(18,41,32,56)(20,44,36,42)(24,50,52,70)(26,54,48,28)(34,61,49,59)(38,62,46,67)(39,55,47,40)(43,66,60,65)(53,71,69,57)(63,72,68,64);(1,4,5,3)(2,8,7,6)(9,13,14,33)(10,18,19,32)(11,24,25,52)(12,29,30,58)(15,34,22,49)(16,38,21,46)(17,41,27,56)(20,43,36,60)(23,50,51,70)(26,53,48,69)(28,57,54,71)(31,59,35,61)(37,62,45,67)(39,63,47,68)(40,64,55,72)(42,65,44,66)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269671);