Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222256 |
| 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 | |
1336 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)(12,17)(13,30,32,18)(14,20)(15,19)(22,47,50,36)(23,37)(24,45,33,38)(26,54,53,39)(27,56,58,40)(28,42)(29,41)(31,59,62,43)(34,46)(35,44)(48,52)(49,63)(51,64)(55,65)(57,68,61,66)(60,67)(69,70)(71,72);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,26)(13,23)(14,33)(15,25)(17,39)(18,37)(19,21)(20,45)(22,28)(24,35)(27,55)(29,53)(30,48)(31,60)(32,52)(34,50)(36,42)(38,44)(40,65)(41,54)(43,67)(46,47)(49,57)(51,62)(56,72)(58,71)(59,64)(61,70)(63,66)(68,69);(2,7)(6,8)(9,12,15,29)(10,17,19,41)(11,22,25,50)(13,27,32,58)(14,34,35,28)(16,36,21,47)(18,40,30,56)(20,46,44,42)(23,49,52,70)(24,53,33,26)(31,61,62,57)(37,63,48,69)(38,54,45,39)(43,68,59,66)(51,71,60,55)(64,72,67,65);(1,4,5,3)(2,8,7,6)(9,13,15,32)(10,18,19,30)(11,23,25,52)(12,27,29,58)(14,31,35,62)(16,37,21,48)(17,40,41,56)(20,43,44,59)(22,49,50,70)(24,51,33,60)(26,55,53,71)(28,57,34,61)(36,63,47,69)(38,64,45,67)(39,65,54,72)(42,66,46,68)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,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, 268477);