#239938

order	512
nilpotency class	3
isoclinism	isoclinic class of #239921
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

922 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)(3,7)(4,6)(5,8)(9,10,15,19)(11,20,23,32)(12,24,28,17)(13,18,14,29)(16,30,33,22)(21,37,25,36)(26,39,27,35)(31,38,34,40);(9,11,15,23)(10,16,19,33)(12,25,28,21)(13,22,14,30)(17,34,24,31)(18,32,29,20)(26,40,27,38)(35,36,39,37);(2,6)(3,4)(7,8)(9,12,14,27,15,28,13,26)(10,17,18,35,19,24,29,39)(11,21,22,38,23,25,30,40)(16,31,20,36,33,34,32,37);(1,3,5,4)(2,7,8,6)(9,13,15,14)(10,18,19,29)(11,22,23,30)(12,26,28,27)(16,32,33,20)(17,35,24,39)(21,38,25,40)(31,37,34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147134);