Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16319 |
| 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 | |
917 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,13,7,26)(3,14,17,32,18,9,27,39)(4,19,28,34,21,48,22,10)(5,11,29,35,24,36,25,53)(15,40,42,37,43,46,54,30)(16,38,44,58,45,31,55,60)(20,49,52,33,23,47,50,59)(41,62,63,56,51,61,64,57);(1,3,8,27,6,18,7,17)(2,9,26,39,13,14,12,32)(4,15,28,54,21,43,22,42)(5,16,29,55,24,45,25,44)(10,30,48,40,34,37,19,46)(11,31,53,60,36,38,35,58)(20,41,52,64,23,51,50,63)(33,56,49,61,59,57,47,62);(1,4,7,22,6,21,8,28)(2,10,12,19,13,34,26,48)(3,15,17,42,18,43,27,54)(5,23,25,52,24,20,29,50)(9,30,32,46,14,37,39,40)(11,33,35,47,36,59,53,49)(16,41,44,63,45,51,55,64)(31,57,58,61,38,56,60,62);(1,5)(2,11)(3,16)(4,20)(6,24)(7,25)(8,29)(9,31)(10,33)(12,35)(13,36)(14,38)(15,41)(17,44)(18,45)(19,47)(21,23)(22,50)(26,53)(27,55)(28,52)(30,56)(32,58)(34,59)(37,57)(39,60)(40,61)(42,63)(43,51)(46,62)(48,49)(54,64)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,d^c=de,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365714);