#6410988

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=g,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6318184);