#7111408

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,36)(28,33)(29,34)(30,44)(31,35)(32,39)(37,51)(38,47)(40,46)(41,48)(42,50)(43,49)(45,53)(52,56)(54,55);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,29,38,31,40,30,39)(26,32,44,46,35,47,34,36)(28,37,43,52,42,54,41,53)(33,45,48,55,50,56,49,51);(2,5)(6,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,31,42)(26,33,35,50)(27,37,40,54)(29,43,30,41)(32,45,47,56)(34,49,44,48)(36,51,46,55)(38,52,39,53);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(27,38)(29,30)(32,46)(34,44)(36,47)(37,52)(39,40)(41,43)(45,55)(48,49)(51,56)(53,54);(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,34,35,44)(27,38,40,39)(28,41,42,43)(32,36,47,46)(33,48,50,49)(37,53,54,52)(45,55,56,51)

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=hi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7018604);