#6404201

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61030	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311397);