Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388379 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14519 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1247 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,20,31,18,9,19,32)(4,10,24,35,23,36,28,41)(5,11,25,39,26,38,27,40)(16,29,21,33,44,42,48,57)(17,30,46,55,45,37,47,56)(22,34,53,60,51,61,52,59)(43,62,49,64,63,54,50,58);(1,3)(2,9)(4,21)(5,17)(6,20)(7,19)(8,18)(10,33)(11,37)(12,15)(13,32)(14,31)(16,24)(22,50)(23,48)(25,47)(26,45)(27,46)(28,44)(29,35)(30,38)(34,58)(36,57)(39,55)(40,56)(41,42)(43,52)(49,51)(53,63)(54,60)(59,62)(61,64);(1,4,7,28,8,23,6,24)(2,10,13,35,12,36,14,41)(3,16,20,48,18,44,19,21)(5,22,25,52,26,51,27,53)(9,29,31,33,15,42,32,57)(11,34,40,60,38,61,39,59)(17,43,46,50,45,63,47,49)(30,54,56,58,37,62,55,64);(1,5,8,26)(2,11,12,38)(3,17,18,45)(4,22,23,51)(6,27,7,25)(9,30,15,37)(10,34,36,61)(13,40,14,39)(16,43,44,63)(19,47,20,46)(21,49,48,50)(24,53,28,52)(29,54,42,62)(31,56,32,55)(33,58,57,64)(35,60,41,59)Polycyclic presentation:
2^9|a^2=h,c^2=eh,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295578);