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 | |
3648 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)(5,6,11,18,12,16,10,17)(7,19,23,36,24,13,22,37)(8,14,27,41,28,40,32,43)(9,29,30,44,31,15,33,42)(20,45,25,53,48,34,51,38)(21,35,49,60,50,46,52,61)(26,55,57,63,58,39,56,64)(47,59,66,68,67,65,54,62);(5,7,12,24)(6,13,16,19)(8,25,28,51)(9,21,31,50)(10,23,11,22)(14,38,40,53)(15,35,29,46)(17,37,18,36)(20,32,48,27)(26,54,58,66)(30,52,33,49)(34,43,45,41)(39,62,55,68)(42,61,44,60)(47,57,67,56)(59,64,65,63);(2,4)(5,8)(6,14)(7,20)(9,26)(10,32)(11,27)(12,28)(13,34)(15,39)(16,40)(17,43)(18,41)(19,45)(21,47)(22,51)(23,25)(24,48)(29,55)(30,57)(31,58)(33,56)(35,59)(36,53)(37,38)(42,64)(44,63)(46,65)(49,66)(50,67)(52,54)(60,68)(61,62);(5,9,12,31)(6,15,16,29)(7,21,24,50)(8,26,28,58)(10,30,11,33)(13,35,19,46)(14,39,40,55)(17,42,18,44)(20,47,48,67)(22,49,23,52)(25,54,51,66)(27,56,32,57)(34,59,45,65)(36,60,37,61)(38,62,53,68)(41,63,43,64)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,g^2=i,h^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, 266812);