Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458569 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16320 |
| 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 | |
951 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)(3,15)(4,10)(5,26,29,41)(6,13)(7,12)(8,14)(9,18)(11,30,40,28)(16,43)(17,47,50,32)(19,34)(20,33)(21,31)(22,39,54,61)(23,42)(24,37)(25,38)(27,36,55,57)(35,46)(44,49,56,51)(45,62,64,58)(48,60,53,63)(52,59);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,21,23,46,25,52,24,16)(5,17,30,51,29,50,28,49)(10,35,37,31,38,43,42,59)(11,32,41,44,40,47,26,56)(22,53,55,64,54,48,27,45)(36,60,61,58,57,63,39,62);(1,4)(2,10)(3,16)(5,27,29,55)(6,23)(7,24)(8,25)(9,31)(11,39,40,61)(12,37)(13,42)(14,38)(15,43)(17,48,50,53)(18,21)(19,52)(20,46)(22,28,54,30)(26,57,41,36)(32,60,47,63)(33,35)(34,59)(44,58,56,62)(45,49,64,51);(1,5)(2,11)(3,17)(4,22)(6,30)(7,28)(8,29)(9,32)(10,36)(12,41)(13,26)(14,40)(15,44)(16,45)(18,51)(19,49)(20,50)(21,53)(23,55)(24,27)(25,54)(31,58)(33,56)(34,47)(35,60)(37,61)(38,57)(39,42)(43,63)(46,64)(48,52)(59,62)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365765);