#7194827

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1155 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7102023);