Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222256 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
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 | |
3662 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)(18,43,46,32)(19,33)(20,41,29,34)(22,50,49,35)(23,52,54,36)(24,38)(25,37)(27,55,58,39)(30,42)(31,40)(44,48)(45,59)(47,60)(51,61)(53,64,57,62)(56,63)(65,66)(67,68);(5,7)(6,12)(8,22)(9,19)(10,29)(11,21)(13,35)(14,33)(15,17)(16,41)(18,24)(20,31)(23,51)(25,49)(26,44)(27,56)(28,48)(30,46)(32,38)(34,40)(36,61)(37,50)(39,63)(42,43)(45,53)(47,58)(52,68)(54,67)(55,60)(57,66)(59,62)(64,65);(2,4)(5,8,11,25)(6,13,15,37)(7,18,21,46)(9,23,28,54)(10,30,31,24)(12,32,17,43)(14,36,26,52)(16,42,40,38)(19,45,48,66)(20,49,29,22)(27,57,58,53)(33,59,44,65)(34,50,41,35)(39,64,55,62)(47,67,56,51)(60,68,63,61);(5,9,11,28)(6,14,15,26)(7,19,21,48)(8,23,25,54)(10,27,31,58)(12,33,17,44)(13,36,37,52)(16,39,40,55)(18,45,46,66)(20,47,29,56)(22,51,49,67)(24,53,30,57)(32,59,43,65)(34,60,41,63)(35,61,50,68)(38,62,42,64)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,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, 266808);