#6399617

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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

2105 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)(9,10,14,18,15,19,13,28)(11,20,23,16,21,32,29,30)(12,17,27,35,25,34,26,40)(22,36,37,31,24,38,39,33);(2,5)(3,7)(11,21)(16,30)(20,32)(22,24)(23,29)(31,33)(36,38)(37,39);(9,11)(10,16)(12,24)(13,29)(14,23)(15,21)(17,33)(18,32)(19,30)(20,28)(22,25)(26,37)(27,39)(31,34)(35,36)(38,40);(1,3,4,7)(2,6,5,8)(12,25)(17,34)(22,24)(26,27)(31,33)(35,40)(36,38)(37,39);(2,5)(6,8)(9,12,14,27,15,25,13,26)(10,17,18,35,19,34,28,40)(11,22,23,37,21,24,29,39)(16,31,32,38,30,33,20,36)

Polycyclic presentation:

2^9|a^2=h,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^c=ei,g^c=gi,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, 6306813);