Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396373 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| 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 | |
1336 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)(5,6,10,17,12,18,11,16)(7,19,22,13,24,37,23,38)(8,25,29,14,30,41,28,42)(9,31,32,44,34,43,33,15)(20,35,48,45,49,53,26,39)(21,36,50,60,52,62,51,46)(27,56,58,54,59,40,57,64)(47,63,66,61,67,65,55,68);(1,2,3,4)(5,7,10,22,12,24,11,23)(6,13,16,19,18,38,17,37)(8,26,28,49,30,48,29,20)(9,21,32,50,34,52,33,51)(14,39,41,35,42,45,25,53)(15,36,43,46,44,62,31,60)(27,55,57,67,59,66,58,47)(40,63,64,61,56,65,54,68);(5,8)(6,14)(7,20)(9,27)(10,28)(11,29)(12,30)(13,35)(15,40)(16,41)(17,25)(18,42)(19,45)(21,47)(22,26)(23,48)(24,49)(31,54)(32,57)(33,58)(34,59)(36,61)(37,39)(38,53)(43,64)(44,56)(46,65)(50,55)(51,66)(52,67)(60,63)(62,68);(5,9,10,32,12,34,11,33)(6,15,16,43,18,44,17,31)(7,21,22,50,24,52,23,51)(8,27,28,57,30,59,29,58)(13,36,19,46,38,62,37,60)(14,40,41,64,42,56,25,54)(20,47,26,55,49,67,48,66)(35,61,45,65,53,68,39,63)Polycyclic presentation:
2^9|a^2=fi,b^2=e,d^2=hi,e^2=i,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,g^b=gi,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, 319639);