Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422131 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×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 | |
1107 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)(7,13)(8,24)(9,30)(10,17)(11,16)(12,18)(14,27)(15,31)(19,39)(20,48)(21,37)(22,36)(23,38)(25,52)(26,55)(28,42)(29,41)(32,43)(33,44)(34,46)(35,49)(40,58)(45,65)(47,60)(50,61)(51,62)(53,56)(54,68)(57,64)(59,67)(63,66);(1,2)(3,4)(5,7,10,21,12,23,11,22)(6,13,17,37,18,38,16,36)(8,25,28,47,29,46,27,19)(9,20,31,49,32,50,33,51)(14,39,24,52,42,60,41,34)(15,35,43,61,44,62,30,48)(26,54,57,67,58,66,56,45)(40,63,53,65,55,68,64,59);(1,2,4,3)(5,8,10,27,12,29,11,28)(6,14,16,24,18,42,17,41)(7,19,21,46,23,47,22,25)(9,26,31,56,32,58,33,57)(13,34,36,39,38,52,37,60)(15,40,30,53,44,55,43,64)(20,45,49,66,50,67,51,54)(35,59,48,63,62,65,61,68);(1,3,4,2)(5,9)(6,15)(7,20)(8,26)(10,31)(11,33)(12,32)(13,35)(14,40)(16,30)(17,43)(18,44)(19,45)(21,49)(22,51)(23,50)(24,53)(25,54)(27,56)(28,57)(29,58)(34,59)(36,48)(37,61)(38,62)(39,63)(41,64)(42,55)(46,66)(47,67)(52,65)(60,68)Polycyclic presentation:
2^9|b^2=eg,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345526);