#6567075

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1232 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,41)(29,34)(30,45)(31,35)(32,40)(33,42)(37,52)(38,51)(39,47)(43,48)(44,49)(46,53)(50,54)(55,56);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,40,31,38,29,39)(26,32,34,36,35,47,45,51)(28,37,42,54,43,53,44,55)(33,46,41,50,49,52,48,56);(1,3,5,4)(2,7,8,6)(11,17)(14,20)(18,23)(21,24)(25,28,30,42,31,43,29,44)(26,33,34,41,35,49,45,48)(27,37,40,54,38,53,39,55)(32,46,36,50,47,52,51,56);(1,4,5,3)(2,6,8,7)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,38)(32,47)(36,51)(37,53)(39,40)(46,52)(50,56)(54,55);(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,45)(27,39,38,40)(28,42,43,44)(32,36,47,51)(33,48,49,41)(37,54,53,55)(46,56,52,50)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^c=ei,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, 6474271);