#7143829

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2216 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)(13,14,20,17)(15,24,28,26)(16,22,31,33)(18,23,19,34)(21,35,32,27)(25,41,29,44)(30,39,36,40)(37,42,38,43);(6,9)(7,11)(13,15,18,35,20,28,19,27)(14,21,34,26,17,32,23,24)(16,29,36,43,31,25,30,42)(22,38,39,41,33,37,40,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,31)(14,22,17,33)(15,25,28,29)(18,36,19,30)(21,37,32,38)(23,40,34,39)(24,41,26,44)(27,43,35,42);(6,9)(10,12)(13,17,20,14)(15,26,28,24)(16,22,31,33)(18,34,19,23)(21,27,32,35)(25,41,29,44)(30,39,36,40)(37,42,38,43);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,17,34)(15,27,28,35)(16,30,31,36)(21,26,32,24)(22,39,33,40)(25,42,29,43)(37,41,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,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, 7051025);