Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2544 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14915 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1307 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)(3,15)(4,10)(5,26)(6,13)(7,12)(8,14)(9,18)(11,29)(16,34)(17,31)(19,33)(20,32)(21,48)(22,50)(23,40)(24,36)(25,37)(27,39)(28,38)(30,43)(35,51)(41,47)(42,61)(44,58)(45,54)(46,57)(49,64)(52,55)(53,60)(56,63)(59,62);(1,3,7,19,8,20,6,18)(2,9,13,32,14,33,12,15)(4,21,23,16,25,43,24,44)(5,17,29,47,28,46,27,45)(10,34,36,48,37,58,40,30)(11,31,26,54,39,57,38,41)(22,49,51,42,52,62,53,63)(35,59,50,61,60,64,55,56);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,30)(11,35)(12,36)(13,40)(14,37)(15,34)(17,42)(18,43)(19,21)(20,44)(26,55)(27,51)(28,52)(29,53)(31,56)(32,58)(33,48)(38,50)(39,60)(41,59)(45,62)(46,63)(47,49)(54,64)(57,61);(1,5,6,27,8,28,7,29)(2,11,12,38,14,39,13,26)(3,17,18,45,20,46,19,47)(4,22,23,51,25,52,24,53)(9,31,15,41,33,57,32,54)(10,35,36,50,37,60,40,55)(16,42,43,62,44,63,21,49)(30,56,34,59,48,61,58,64)Polycyclic presentation:
2^9|b^2=h,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312804);