#240162

order	512
nilpotency class	3
isoclinism	isoclinic class of #240158
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
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

929 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=gh,d^c=dh,f^c=fhi

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, 147358);