Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #390401 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14571 |
| 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 | |
1022 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)(3,13,17,39)(4,18,22,9)(5,23)(6,11)(7,12)(8,16,32,27)(10,25)(14,43)(15,46,47,31)(19,41)(20,53,56,59)(21,50,58,35)(24,42,48,36)(26,37)(28,38)(29,40)(30,45)(33,52)(34,55,49,57)(44,61)(51,62)(54,63)(60,64);(1,3)(2,8)(4,19)(5,24)(6,27)(7,17)(9,33)(10,36)(11,39)(12,32)(13,40)(14,22)(15,25)(16,29)(18,30)(20,54)(21,45)(23,46)(26,48)(28,47)(31,37)(34,63)(35,41)(38,42)(43,50)(44,57)(49,64)(51,55)(52,58)(53,62)(56,60)(59,61);(1,4)(2,9)(3,14,17,19)(5,20)(6,21)(7,22)(8,30,32,33)(10,34)(11,35)(12,18)(13,41,39,43)(15,44,47,51)(16,45,27,52)(23,59)(24,60,48,54)(25,55)(26,56)(28,57)(29,58)(31,61,46,62)(36,64,42,63)(37,53)(38,49)(40,50);(1,5,7,26)(2,10,12,38)(3,15,17,47)(4,20,22,56)(6,28,29,25)(8,31,32,46)(9,34,18,49)(11,23,40,37)(13,42,39,36)(14,44,19,51)(16,48,27,24)(21,57,58,55)(30,61,33,62)(35,59,50,53)(41,63,43,64)(45,54,52,60)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297598);