Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1154 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)(7,17,21,42)(8,13)(9,14)(10,15)(11,16)(12,20,33,29)(18,44,48,59)(19,45,51,66)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(30,43)(31,52,34,47)(32,50,61,57)(46,67,68,60)(49,58,55,62)(53,63)(54,64)(56,65);(5,7)(6,12)(8,18)(9,19)(10,29)(11,21)(13,34)(14,32)(15,42)(16,33)(17,39)(20,26)(22,46)(23,52)(24,48)(25,55)(27,57)(28,51)(30,47)(31,37)(35,62)(36,59)(38,60)(40,66)(41,61)(43,44)(45,63)(49,56)(50,53)(54,68)(58,64)(65,67);(2,4)(5,8,11,24)(6,13,16,37)(7,18,21,48)(9,25,28,56)(10,30,26,23)(12,31,33,34)(14,38,41,65)(15,43,39,36)(17,44,42,59)(19,49,51,55)(20,52,29,47)(22,53,54,27)(32,60,61,67)(35,63,64,40)(45,58,66,62)(46,57,68,50);(5,9,26,53,11,28,10,27)(6,14,39,63,16,41,15,40)(7,19,20,50,21,51,29,57)(8,22,30,56,24,54,23,25)(12,32,17,45,33,61,42,66)(13,35,43,65,37,64,36,38)(18,46,47,49,48,68,52,55)(31,58,59,60,34,62,44,67)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=efhi,e^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289082);