Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| 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 | |
1634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6,11,16,12,18,10,17)(7,19,23,37,24,43,22,13)(8,25,28,14,30,39,29,40)(9,31,33,42,34,41,32,15)(20,35,47,61,26,54,52,44)(21,48,50,63,51,36,49,45)(27,56,57,53,59,38,58,62)(46,64,65,60,55,68,67,66);(1,2,4,3)(5,7,10,22,12,24,11,23)(6,13,16,19,18,37,17,43)(8,26,28,52,30,20,29,47)(9,21,32,49,34,51,33,50)(14,35,25,44,40,54,39,61)(15,36,31,45,42,48,41,63)(27,55,57,67,59,46,58,65)(38,60,53,64,56,66,62,68);(5,8)(6,14)(7,20,24,26)(9,27)(10,28)(11,29)(12,30)(13,35,37,54)(15,38)(16,25)(17,39)(18,40)(19,44,43,61)(21,46,51,55)(22,47,23,52)(31,53)(32,57)(33,58)(34,59)(36,60,48,66)(41,62)(42,56)(45,64,63,68)(49,65,50,67);(1,3,4,2)(5,9,11,33,12,34,10,32)(6,15,17,41,18,42,16,31)(7,21,23,50,24,51,22,49)(8,27,29,58,30,59,28,57)(13,36,43,63,37,48,19,45)(14,38,39,62,40,56,25,53)(20,46,52,67,26,55,47,65)(35,60,61,68,54,66,44,64)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281636);