Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448851 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15922 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
727 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,29)(3,9,16,33,17,34,30,35)(4,18,21,38,23,41,22,10)(5,24,28,11,26,42,27,40)(14,43,45,36,19,50,48,31)(15,46,49,39,25,56,47,32)(20,37,55,51,53,59,54,60)(44,58,63,64,52,57,62,61);(1,3,6,16,8,17,7,30)(2,9,12,33,13,34,29,35)(4,19,21,48,23,14,22,45)(5,25,28,47,26,15,27,49)(10,36,18,50,38,31,41,43)(11,39,42,56,40,32,24,46)(20,44,55,63,53,52,54,62)(37,57,51,61,59,58,60,64);(1,4,8,23)(2,10,13,38)(3,14,17,19)(5,20,26,53)(6,22,7,21)(9,31,34,36)(11,37,40,59)(12,41,29,18)(15,44,25,52)(16,48,30,45)(24,51,42,60)(27,55,28,54)(32,57,39,58)(33,50,35,43)(46,64,56,61)(47,63,49,62);(1,5)(2,11)(3,15,17,25)(4,20)(6,28)(7,27)(8,26)(9,32,34,39)(10,37)(12,42)(13,40)(14,44,19,52)(16,49,30,47)(18,51)(21,55)(22,54)(23,53)(24,29)(31,57,36,58)(33,46,35,56)(38,59)(41,60)(43,61,50,64)(45,63,48,62)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,d^2=g,f^2=i,h^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,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356048);