#6376322

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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	#3449	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1891 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)(9,10,15,20,16,21,14,32)(11,17,25,36,23,35,24,39)(12,26,22,34,28,40,27,18)(13,19,31,38,29,37,30,33);(1,2)(3,6)(4,5)(7,8)(9,11,15,25,16,23,14,24)(10,17,20,36,21,35,32,39)(12,13,22,31,28,29,27,30)(18,33,26,19,34,38,40,37);(2,5)(6,8)(9,12,14,27,16,28,15,22)(10,18,20,26,21,34,32,40)(11,13,24,30,23,29,25,31)(17,33,36,19,35,38,39,37);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,22)(12,24)(14,30)(15,31)(16,29)(17,34)(18,35)(20,38)(21,37)(23,27)(25,28)(26,39)(32,33)(36,40);(11,23)(13,29)(17,35)(19,37)(24,25)(30,31)(33,38)(36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^b=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, 6283518);