Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #402446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| 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 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,3,4)(5,6)(7,17,20,12)(8,22)(9,27)(10,15)(11,31)(13,25)(14,30)(16,28)(18,35,23,32)(19,43,45,55)(21,34,44,40)(24,52)(26,37)(29,57)(33,46,39,64)(36,63)(38,58)(41,65,50,67)(42,60,51,62)(47,54)(48,49)(53,56)(59,66,61,68);(5,7,11,21,10,20,28,44)(6,12,31,40,15,17,16,34)(8,23,49,42,25,18,26,51)(9,19,30,46,29,45,58,64)(13,35,37,62,22,32,48,60)(14,33,57,43,38,39,27,55)(24,50,63,66,53,41,54,68)(36,61,56,65,47,59,52,67);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,32)(14,36)(15,22)(16,37)(17,35)(19,41)(20,23)(21,42)(27,56)(28,49)(29,53)(30,54)(31,48)(33,59)(34,60)(38,47)(39,61)(40,62)(43,65)(44,51)(45,50)(46,66)(52,57)(55,67)(58,63)(64,68);(5,9,28,58,10,29,11,30)(6,14,31,57,15,38,16,27)(7,19,44,64,20,45,21,46)(8,24,49,63,25,53,26,54)(12,33,40,43,17,39,34,55)(13,36,48,52,22,47,37,56)(18,41,51,68,23,50,42,66)(32,59,62,65,35,61,60,67)Polycyclic presentation:
2^9|a^2=fi,b^2=eh,d^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323406);