Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #444648 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15784 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,27,33)(3,13,16,39,17,8,26,30)(4,18,22,47,21,9,31,35)(5,23,24,52,25,10,36,38)(14,28,19,32,42,40,49,45)(15,37,43,61,44,29,53,55)(20,34,51,59,50,46,56,63)(41,54,48,57,62,60,64,58);(1,3)(2,8)(4,19)(5,15)(6,26)(7,17)(9,32)(10,37)(11,13)(12,39)(14,22)(16,27)(18,45)(20,48)(21,49)(23,29)(24,53)(25,44)(28,35)(30,33)(31,42)(34,58)(36,43)(38,55)(40,47)(41,51)(46,57)(50,64)(52,61)(54,63)(56,62)(59,60);(1,4,6,22,7,21,27,31)(2,9,33,47,11,18,12,35)(3,14,16,19,17,42,26,49)(5,20,24,51,25,50,36,56)(8,28,39,45,13,40,30,32)(10,34,52,63,23,46,38,59)(15,41,43,48,44,62,53,64)(29,54,61,58,37,60,55,57);(1,5,7,25)(2,10)(3,15,17,44)(4,20,21,50)(6,24,27,36)(8,29)(9,34)(11,23)(12,38)(13,37)(14,41,42,62)(16,43,26,53)(18,46)(19,48,49,64)(22,51,31,56)(28,54)(30,55)(32,57)(33,52)(35,59)(39,61)(40,60)(45,58)(47,63)Polycyclic presentation:
2^9|a^2=g,c^2=eg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351845);