#6569129

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,5,6)(3,7,4,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,30,35)(27,36,41,50)(28,42,43,49)(29,33,37,34)(31,39,46,40)(32,47,48,44)(38,52,53,45)(51,54,56,55);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,37,39,30,41,29,40)(26,31,33,50,35,46,34,36)(28,38,47,54,43,53,44,55)(32,45,49,56,48,52,42,51);(1,3,5,4)(2,7,6,8)(11,17)(14,20)(18,23)(21,24)(25,28,37,47,30,43,29,44)(26,32,33,49,35,48,34,42)(27,38,39,54,41,53,40,55)(31,45,50,56,46,52,36,51);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(28,43)(32,48)(38,53)(42,49)(44,47)(45,52)(51,56)(54,55);(1,4,5,3)(2,8,6,7)(12,19)(15,22)(18,23)(21,24)(25,29,30,37)(26,33,35,34)(27,39,41,40)(28,44,43,47)(31,36,46,50)(32,49,48,42)(38,54,53,55)(45,51,52,56)

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fi,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6476325);