#7143751

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

2245 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)(15,23,26,37)(16,20,27,34)(17,29,30,21)(19,31,33,25)(24,36,39,44)(28,41,40,35)(32,42,43,38);(6,9)(7,11)(13,15,17,25,18,26,30,31)(14,19,21,23,22,33,29,37)(16,24,28,38,27,39,40,42)(20,32,35,36,34,43,41,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,24)(17,28)(18,27)(19,32)(21,35)(22,34)(23,36)(25,38)(26,39)(29,41)(30,40)(31,42)(33,43)(37,44);(6,9)(10,12)(15,25)(17,30)(19,23)(21,29)(24,38)(26,31)(28,40)(32,36)(33,37)(35,41)(39,42)(43,44);(5,7,8,11)(6,10,9,12)(16,27)(20,34)(24,39)(28,40)(32,43)(35,41)(36,44)(38,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7050947);