Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478528 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17152 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
679 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)(3,15,19,9)(4,20,24,43)(5,11,27,39)(6,13)(7,14)(8,12)(10,23,36,30)(16,34)(17,32)(18,42,29,33)(21,44)(22,51)(25,57)(26,40,28,41)(31,47)(35,54)(37,49)(38,55)(45,50)(46,63,53,61)(48,59)(52,58,64,60)(56,62);(1,3,6,18,8,19,7,29)(2,9,13,33,12,15,14,42)(4,21,23,16,24,47,30,48)(5,25,26,50,27,49,28,17)(10,34,20,31,36,59,43,44)(11,37,40,32,39,57,41,45)(22,53,54,64,56,46,55,52)(35,58,62,61,38,60,51,63);(1,4)(2,10,12,36)(3,16)(5,22)(6,23)(7,30)(8,24)(9,31,15,44)(11,38,39,35)(13,20,14,43)(17,46)(18,47)(19,48)(21,29)(25,52)(26,54)(27,56)(28,55)(32,60,45,58)(33,59,42,34)(37,61,57,63)(40,51,41,62)(49,64)(50,53);(1,5,8,27)(2,11)(3,17,19,50)(4,22,24,56)(6,28,7,26)(9,32)(10,35)(12,39)(13,41)(14,40)(15,45)(16,46,48,53)(18,49,29,25)(20,51)(21,52,47,64)(23,55,30,54)(31,58)(33,37)(34,61)(36,38)(42,57)(43,62)(44,60)(59,63)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,g^c=gi,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385725);