Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461151 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16381 |
| 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 | |
843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,31,9,19,34,18,35)(4,20,23,39,25,38,24,10)(5,26,28,11,30,41,29,42)(16,32,21,36,46,61,45,60)(17,47,49,43,51,33,50,57)(22,54,55,40,56,37,27,52)(44,63,53,64,58,62,48,59);(1,3,7,18,8,19,6,31)(2,9,13,15,14,35,12,34)(4,21,23,16,25,45,24,46)(5,17,28,50,30,51,29,49)(10,36,20,32,39,60,38,61)(11,33,41,43,42,47,26,57)(22,53,55,44,56,48,27,58)(37,63,52,59,54,62,40,64);(1,4,6,23,8,25,7,24)(2,10,12,20,14,39,13,38)(3,16,18,21,19,46,31,45)(5,27,28,56,30,55,29,22)(9,32,15,36,35,61,34,60)(11,40,41,54,42,52,26,37)(17,48,49,58,51,53,50,44)(33,62,57,64,47,63,43,59);(1,5,8,30)(2,11,14,42)(3,17,19,51)(4,22,25,56)(6,29,7,28)(9,33,35,47)(10,37,39,54)(12,26,13,41)(15,43,34,57)(16,44,46,58)(18,50,31,49)(20,52,38,40)(21,53,45,48)(23,55,24,27)(32,59,61,64)(36,63,60,62)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,e^d=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368348);