#6396566

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	#60797	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,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, 6303762);