#7110918

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996335
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	#59110	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1539 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,29,32,30,34,27,31)(28,33,36,39,37,40,35,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(27,29)(28,36)(32,34)(35,37)(38,39);(2,6)(5,7)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27,30,29)(26,32,34,31)(28,36,37,35)(33,38,40,39);(3,8)(5,7)(12,20)(15,24)(17,19)(21,23)(25,28,29,36,30,37,27,35)(26,33,32,39,34,40,31,38);(10,16)(11,18)(15,24)(17,19)(25,27,30,29)(26,31,34,32)(28,35,37,36)(33,38,40,39)

Polycyclic presentation:

2^9|a^2=fgh,c^2=h,d^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7018114);