Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219706 |
| 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 | |
1147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16,12,17,10,34)(7,18,22,37,23,48,21,13)(8,14,28,40,27,41,26,45)(9,29,32,15,33,43,31,44)(19,46,51,35,50,38,24,55)(20,52,53,47,54,36,30,42)(25,57,60,63,59,64,58,39)(49,66,56,62,68,61,67,65);(1,2)(3,4)(5,7,10,21,12,23,11,22)(6,13,16,18,17,37,34,48)(8,24,26,50,27,51,28,19)(9,30,31,54,33,53,32,20)(14,38,40,55,41,46,45,35)(15,42,43,52,44,47,29,36)(25,49,58,67,59,68,60,56)(39,61,57,65,63,66,64,62);(2,4)(5,8)(6,14,17,41)(7,19)(9,25)(10,28)(11,26)(12,27)(13,35,37,55)(15,39,44,63)(16,45,34,40)(18,46,48,38)(20,49)(21,51)(22,24)(23,50)(29,57,43,64)(30,56)(31,60)(32,58)(33,59)(36,61,52,66)(42,62,47,65)(53,67)(54,68);(5,9,12,33)(6,15,17,44)(7,20,23,54)(8,25,27,59)(10,32,11,31)(13,36,37,52)(14,39,41,63)(16,29,34,43)(18,47,48,42)(19,49,50,68)(21,53,22,30)(24,56,51,67)(26,60,28,58)(35,61,55,66)(38,62,46,65)(40,64,45,57)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295659);