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 | |
3460 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)(7,19,22,43,23,13,32,36)(8,14,26,40,27,39,31,42)(9,28,29,44,30,15,33,41)(20,45,48,37,49,34,24,53)(21,35,51,61,52,46,50,60)(25,55,57,63,58,38,56,64)(47,59,54,62,66,65,67,68);(5,7,10,22,12,23,11,32)(6,13,17,36,16,19,18,43)(8,24,26,49,27,48,31,20)(9,21,29,50,30,52,33,51)(14,37,40,45,39,53,42,34)(15,35,41,60,28,46,44,61)(25,54,56,66,58,67,57,47)(38,62,63,65,55,68,64,59);(2,4)(5,8,12,27)(6,14,16,39)(7,20,23,49)(9,25,30,58)(10,31,11,26)(13,34,19,45)(15,38,28,55)(17,42,18,40)(21,47,52,66)(22,48,32,24)(29,57,33,56)(35,59,46,65)(36,53,43,37)(41,64,44,63)(50,67,51,54)(60,68,61,62);(5,9)(6,15)(7,21)(8,25)(10,29)(11,33)(12,30)(13,35)(14,38)(16,28)(17,41)(18,44)(19,46)(20,47)(22,50)(23,52)(24,54)(26,56)(27,58)(31,57)(32,51)(34,59)(36,60)(37,62)(39,55)(40,63)(42,64)(43,61)(45,65)(48,67)(49,66)(53,68)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,g^2=i,h^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, 268837);