#359329

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=eh,e^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,h^b=hi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266525);