#237154

order	512
nilpotency class	3
isoclinism	isoclinic class of #237115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144350);