Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
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×#578 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6)(7,17,21,12)(8,13)(9,26,28,14)(10,16)(11,15)(18,41,45,31)(19,32)(20,43,49,33)(22,50,48,34)(23,52,54,35)(24,37)(25,36)(27,55,57,38)(29,40)(30,39)(42,47)(44,58)(46,59)(51,60)(53,62,56,61)(63,65)(64,67)(66,68);(5,7,10,20,11,21,30,49)(6,12,16,33,15,17,39,43)(8,22,24,45,25,48,29,18)(9,19,27,46,28,47,57,67)(13,34,37,41,36,50,40,31)(14,32,38,59,26,42,55,64)(23,51,53,65,54,66,56,44)(35,60,61,63,52,68,62,58);(2,4)(5,8,11,25)(6,13,15,36)(7,18,21,45)(9,23,28,54)(10,29,30,24)(12,31,17,41)(14,35,26,52)(16,40,39,37)(19,44,47,65)(20,48,49,22)(27,56,57,53)(32,58,42,63)(33,50,43,34)(38,62,55,61)(46,66,67,51)(59,68,64,60);(5,9)(6,14)(7,19)(8,23)(10,27)(11,28)(12,32)(13,35)(15,26)(16,38)(17,42)(18,44)(20,46)(21,47)(22,51)(24,53)(25,54)(29,56)(30,57)(31,58)(33,59)(34,60)(36,52)(37,61)(39,55)(40,62)(41,63)(43,64)(45,65)(48,66)(49,67)(50,68)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,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, 268834);