#7187878

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994171
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=gi,e^2=hi,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,f^d=fi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7095074);