Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #435393 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15526 |
| 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 | |
946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,31,8,13,6,12)(3,14,17,34,19,43,18,9)(4,20,23,10,24,36,29,37)(5,25,30,39,28,11,27,40)(15,44,48,59,21,32,46,41)(16,42,26,33,47,38,49,60)(22,52,54,35,56,50,55,57)(45,63,53,61,51,58,64,62);(1,3)(2,9,13,34)(4,21,24,15)(5,26,28,49)(6,17)(7,18)(8,19)(10,32)(11,38)(12,14,31,43)(16,27,47,30)(20,41)(22,53)(23,48,29,46)(25,42)(33,40)(35,61,57,62)(36,59)(37,44)(39,60)(45,54)(50,63,52,58)(51,55)(56,64);(1,4,7,23,8,24,6,29)(2,10,12,20,13,37,31,36)(3,15,18,46,19,21,17,48)(5,22,27,55,28,56,30,54)(9,32,14,41,34,44,43,59)(11,35,40,50,25,57,39,52)(16,45,26,53,47,51,49,64)(33,58,42,61,60,63,38,62);(1,5)(2,11)(3,16)(4,22)(6,30)(7,27)(8,28)(9,33)(10,35)(12,40)(13,25)(14,42)(15,45)(17,49)(18,26)(19,47)(20,50)(21,51)(23,55)(24,56)(29,54)(31,39)(32,58)(34,60)(36,52)(37,57)(38,43)(41,61)(44,63)(46,53)(48,64)(59,62)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,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, 342589);