#7111498

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1350 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)(27,38)(28,34)(29,35)(30,36)(31,47)(32,37)(33,41)(39,49)(40,48)(42,50)(43,52)(44,56)(45,53)(46,51)(54,55);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,41,32,42,30,46)(26,33,36,38,37,51,47,50)(28,39,43,40,29,44,45,55)(34,48,53,49,35,54,52,56);(2,5)(6,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,32,29)(26,34,37,35)(27,39,42,44)(30,45,31,43)(33,48,51,54)(36,53,47,52)(38,49,50,56)(40,46,55,41);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,29,32,28)(26,35,37,34)(27,40,42,55)(30,45,31,43)(33,49,51,56)(36,53,47,52)(38,48,50,54)(39,46,44,41);(10,16)(11,18)(15,24)(17,19)(25,30,32,31)(26,36,37,47)(27,41,42,46)(28,43,29,45)(33,50,51,38)(34,52,35,53)(39,55,44,40)(48,49,54,56)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=hi,d^2=gi,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=ehi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7018694);