Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #451917 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16021 |
| 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 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,27,7,13,6,12)(3,14,26,9,18,38,17,32)(4,19,21,47,22,10,28,34)(5,11,24,36,25,37,29,54)(15,40,42,30,43,39,51,45)(16,35,23,52,44,59,53,31)(20,46,55,58,50,33,49,63)(41,60,64,62,61,57,48,56);(1,3,7,18)(2,9,13,32)(4,15,22,43)(5,23,25,53)(6,26,8,17)(10,30,19,45)(11,35,37,59)(12,38,27,14)(16,29,44,24)(20,48,50,64)(21,51,28,42)(31,54,52,36)(33,57,46,60)(34,40,47,39)(41,55,61,49)(56,63,62,58);(1,4,6,21,7,22,8,28)(2,10,12,34,13,19,27,47)(3,15,17,42,18,43,26,51)(5,20,24,49,25,50,29,55)(9,30,14,39,32,45,38,40)(11,33,36,58,37,46,54,63)(16,41,23,48,44,61,53,64)(31,56,35,57,52,62,59,60);(1,5,6,24,7,25,8,29)(2,11,27,54,13,37,12,36)(3,16,17,23,18,44,26,53)(4,20,21,49,22,50,28,55)(9,31,38,59,32,52,14,35)(10,33,47,63,19,46,34,58)(15,41,42,48,43,61,51,64)(30,56,40,60,45,62,39,57)Polycyclic presentation:
2^9|a^2=efgh,b^2=i,c^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359114);