Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459757 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16348 |
| 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 | |
891 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)(14,43,45,36,19,52,50,31)(15,46,51,60,48,56,49,32)(20,37,55,53,25,39,54,57)(44,59,63,64,47,58,62,61);(1,3,6,16,8,17,7,30)(2,9,12,33,13,34,29,35)(4,19,21,50,23,14,22,45)(5,15,28,51,26,48,27,49)(10,36,18,52,38,31,41,43)(11,32,42,46,40,60,24,56)(20,47,55,62,25,44,54,63)(37,59,53,64,39,58,57,61);(1,4,8,23)(2,10,13,38)(3,14,17,19)(5,25,26,20)(6,22,7,21)(9,31,34,36)(11,39,40,37)(12,41,29,18)(15,47,48,44)(16,50,30,45)(24,57,42,53)(27,54,28,55)(32,59,60,58)(33,52,35,43)(46,61,56,64)(49,62,51,63);(1,5,6,28,8,26,7,27)(2,11,12,42,13,40,29,24)(3,15,16,51,17,48,30,49)(4,20,21,55,23,25,22,54)(9,32,33,46,34,60,35,56)(10,37,18,53,38,39,41,57)(14,44,45,63,19,47,50,62)(31,58,43,61,36,59,52,64)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366954);