Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #364101 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13959 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1168 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)(3,14,19,41)(4,10)(5,25,27,56)(6,12)(7,29)(8,13)(9,17,32,18)(11,26,37,28)(15,39,44,33)(16,45)(20,49,43,30)(21,51,53,55)(22,35)(23,38)(24,36)(31,47)(34,52,59,54)(40,48)(42,57)(46,61)(50,60)(58,62)(63,64);(1,3,7,18,8,19,6,17)(2,9,12,14,13,32,29,41)(4,20,23,44,24,43,22,15)(5,16,28,48,27,47,26,46)(10,33,35,49,36,39,38,30)(11,31,25,40,37,45,56,61)(21,50,54,63,53,62,52,42)(34,58,51,64,59,60,55,57);(1,4,8,24)(2,10,13,36)(3,15,19,44)(5,21,27,53)(6,23,7,22)(9,30,32,49)(11,34,37,59)(12,38,29,35)(14,39,41,33)(16,42,47,63)(17,20,18,43)(25,55,56,51)(26,54,28,52)(31,57,45,64)(40,60,61,58)(46,50,48,62);(1,5)(2,11,13,37)(3,16)(4,21)(6,26)(7,28)(8,27)(9,31,32,45)(10,34,36,59)(12,25,29,56)(14,40,41,61)(15,42)(17,46)(18,48)(19,47)(20,50)(22,52)(23,54)(24,53)(30,57,49,64)(33,58,39,60)(35,51,38,55)(43,62)(44,63)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271298);