Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #433923 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15493 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,31)(3,14,18,35,19,34,17,9)(4,20,23,39,25,38,24,10)(5,26,29,42,30,11,28,41)(15,32,21,36,45,49,48,58)(16,40,46,59,47,43,27,33)(22,52,53,56,54,50,55,37)(44,61,63,60,64,57,51,62);(1,3,6,17,8,19,7,18)(2,9,12,14,13,35,31,34)(4,21,23,45,25,48,24,15)(5,27,28,47,30,46,29,16)(10,36,38,32,39,58,20,49)(11,40,41,59,26,43,42,33)(22,44,53,63,54,64,55,51)(37,61,50,62,56,57,52,60);(1,4,6,24,8,25,7,23)(2,10,12,20,13,39,31,38)(3,15,18,21,19,45,17,48)(5,22,29,53,30,54,28,55)(9,32,34,58,35,49,14,36)(11,37,42,50,26,56,41,52)(16,44,27,51,47,64,46,63)(33,57,40,62,59,61,43,60);(1,5,8,30)(2,11,13,26)(3,16,19,47)(4,22,25,54)(6,29,7,28)(9,33,35,59)(10,37,39,56)(12,42,31,41)(14,43,34,40)(15,44,45,64)(17,46,18,27)(20,50,38,52)(21,51,48,63)(23,55,24,53)(32,57,49,61)(36,60,58,62)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,e^d=ei,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341120);