Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448810 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15921 |
| 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 | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,28,6,12,7,13)(3,14,29,33,17,9,18,40)(4,19,27,10,22,38,23,36)(5,24,25,54,26,11,30,37)(15,41,46,34,20,48,43,31)(16,39,44,60,45,32,56,58)(21,50,51,35,52,47,55,53)(42,61,62,57,49,64,63,59);(1,3,7,18,6,17,8,29)(2,9,13,33,12,14,28,40)(4,20,23,46,22,15,27,43)(5,16,25,44,26,45,30,56)(10,34,19,41,36,31,38,48)(11,32,37,58,24,39,54,60)(21,49,51,63,52,42,55,62)(35,59,47,61,53,57,50,64);(1,4,6,22)(2,10,12,36)(3,15,17,20)(5,21,26,52)(7,27,8,23)(9,31,14,34)(11,35,24,53)(13,38,28,19)(16,42,45,49)(18,46,29,43)(25,55,30,51)(32,57,39,59)(33,41,40,48)(37,50,54,47)(44,63,56,62)(58,61,60,64);(1,5)(2,11)(3,16,17,45)(4,21)(6,26)(7,25)(8,30)(9,32,14,39)(10,35)(12,24)(13,37)(15,42,20,49)(18,44,29,56)(19,47)(22,52)(23,51)(27,55)(28,54)(31,57,34,59)(33,58,40,60)(36,53)(38,50)(41,61,48,64)(43,62,46,63)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=i,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356007);