#236563

order	512
nilpotency class	3
isoclinism	isoclinic class of #236557
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	#60946	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1267 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)(3,5)(4,7)(6,8)(9,10,14,18)(11,15,20,30)(12,22,23,34)(13,25,28,17)(16,27,32,24)(19,35,36,39)(21,33,26,31)(29,38,40,37);(2,5)(3,8)(6,7)(9,11)(10,15)(12,19)(13,26)(14,20)(16,29)(17,33)(18,30)(21,28)(22,39)(23,36)(24,38)(25,31)(27,37)(32,40)(34,35);(1,2)(3,6)(4,7)(5,8)(9,12)(10,16)(11,19)(13,27)(14,23)(15,29)(17,34)(18,32)(20,36)(21,38)(22,25)(24,28)(26,37)(30,40)(31,39)(33,35);(11,20)(12,23)(15,30)(16,32)(21,26)(22,34)(24,27)(31,33)

Polycyclic presentation:

2^9|a^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,d^c=di,e^c=ei,f^c=fi,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, 143759);