Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16267 |
| 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 | |
883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,14,7,13)(3,15,18,34,19,40,30,9)(4,20,23,10,25,36,24,37)(5,26,28,41,29,11,31,39)(16,32,45,51,21,52,50,42)(17,43,47,33,49,60,48,57)(22,54,27,35,55,38,56,61)(44,58,46,59,53,63,64,62);(1,3,7,18,8,19,6,30)(2,9,13,15,14,34,12,40)(4,21,24,50,25,16,23,45)(5,17,28,48,29,49,31,47)(10,32,20,51,37,52,36,42)(11,33,39,43,26,57,41,60)(22,53,27,46,55,44,56,64)(35,58,38,62,61,63,54,59);(1,4,8,25)(2,10,14,37)(3,16,19,21)(5,27,29,56)(6,24,7,23)(9,32,34,52)(11,38,26,54)(12,20,13,36)(15,42,40,51)(17,46,49,64)(18,50,30,45)(22,31,55,28)(33,59,57,62)(35,41,61,39)(43,63,60,58)(44,48,53,47);(1,5)(2,11)(3,17)(4,22)(6,31)(7,28)(8,29)(9,33)(10,35)(12,41)(13,39)(14,26)(15,43)(16,44)(18,48)(19,49)(20,38)(21,53)(23,56)(24,27)(25,55)(30,47)(32,58)(34,57)(36,54)(37,61)(40,60)(42,59)(45,64)(46,50)(51,62)(52,63)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362841);