#7115930

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
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

1263 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(26,31)(28,29)(32,37)(33,39)(34,40)(36,38);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,26,29,31,30,32,28,37)(27,34,36,40,35,39,38,33);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,32)(27,35)(31,37)(36,38);(12,20)(15,24)(17,19)(21,23)(25,27,30,35)(26,33,32,40)(28,36,29,38)(31,39,37,34);(1,4,5,3)(2,6,7,8)(10,16)(11,18)(15,24)(17,19)(25,28,30,29)(26,31,32,37)(27,36,35,38)(33,39,40,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=ghi,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7023126);