Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400051 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,11,18,12,16,10,17)(7,19,23,37,24,13,22,44)(8,25,30,14,28,40,29,41)(9,31,33,42,34,43,32,15)(20,47,54,45,49,38,26,35)(21,50,52,46,53,61,51,36)(27,57,60,55,58,39,59,64)(48,65,68,63,67,62,56,66);(1,2,3,4)(5,7,10,23,12,24,11,22)(6,13,17,44,16,19,18,37)(8,26,29,20,28,54,30,49)(9,21,32,52,34,53,33,51)(14,38,25,35,41,47,40,45)(15,36,43,50,42,46,31,61)(27,56,59,48,58,68,60,67)(39,63,55,62,57,66,64,65);(5,8)(6,14)(7,20)(9,27)(10,30)(11,29)(12,28)(13,35)(15,39)(16,41)(17,40)(18,25)(19,45)(21,48)(22,54)(23,26)(24,49)(31,55)(32,60)(33,59)(34,58)(36,62)(37,47)(38,44)(42,57)(43,64)(46,65)(50,63)(51,68)(52,56)(53,67)(61,66);(5,9,11,33,12,34,10,32)(6,15,17,43,16,42,18,31)(7,21,22,51,24,53,23,52)(8,27,29,59,28,58,30,60)(13,36,44,50,19,46,37,61)(14,39,40,64,41,57,25,55)(20,48,54,68,49,67,26,56)(35,62,38,63,45,65,47,66)Polycyclic presentation:
2^9|a^2=fi,b^2=eh,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323409);