#7266807

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234309
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	#39220	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1357 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,34)(28,32)(29,38)(30,33)(31,36)(35,39)(37,40);(2,5)(3,7)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,36,30,37,28,35)(26,31,32,34,33,39,38,40);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,34,39,40);(2,5)(6,8)(11,18)(14,22)(17,19)(21,23)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,34,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7174003);