#7143142

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7050338);