#236938

order	512
nilpotency class	3
isoclinism	isoclinic class of #236935
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1499 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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,e^b=eh,f^b=fhi,g^b=gh,d^c=dh,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, 144134);