#7188028

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1251 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)(27,34)(28,32)(29,39)(30,33)(31,37)(35,38)(36,40);(9,11)(10,15)(12,14)(13,20)(16,18)(17,24)(19,22)(21,23)(25,27,28,38,30,36,29,37)(26,31,39,40,33,35,32,34);(1,3,4,8)(2,5,6,7)(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,26)(27,35)(28,39)(29,32)(30,33)(31,36)(34,38)(37,40);(11,12)(15,16)(19,20)(23,24)(25,28)(26,32)(29,30)(31,35)(33,39)(37,38);(9,13,14,22)(10,17,18,21)(11,19,12,20)(15,23,16,24)(26,33)(27,36)(32,39)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=g,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,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, 7095224);