Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219542 |
| 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:
(5,6,11,16,12,18,10,17)(7,19,23,38,24,37,22,13)(8,25,29,41,30,46,28,14)(9,15,33,43,34,45,32,44)(20,35,50,39,51,54,26,47)(21,48,52,36,53,42,31,61)(27,57,59,64,60,55,58,40)(49,66,56,63,68,62,67,65);(1,2)(3,4)(5,7,10,22,12,24,11,23)(6,13,16,19,18,38,17,37)(8,26,28,51,30,50,29,20)(9,31,32,53,34,52,33,21)(14,39,25,54,41,47,46,35)(15,42,43,61,45,48,44,36)(27,49,58,67,60,68,59,56)(40,62,57,65,64,66,55,63);(2,4)(5,8)(6,14,18,41)(7,20)(9,27)(10,29)(11,28)(12,30)(13,35,38,54)(15,40,45,64)(16,46,17,25)(19,47,37,39)(21,49)(22,50)(23,26)(24,51)(31,56)(32,59)(33,58)(34,60)(36,62,61,66)(42,63,48,65)(43,55,44,57)(52,67)(53,68);(5,9,12,34)(6,15,18,45)(7,21,24,53)(8,27,30,60)(10,33,11,32)(13,36,38,61)(14,40,41,64)(16,44,17,43)(19,48,37,42)(20,49,51,68)(22,52,23,31)(25,55,46,57)(26,56,50,67)(28,59,29,58)(35,62,54,66)(39,63,47,65)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 295649);