#236576

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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1248 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)(11,16)(12,24)(13,18)(14,28)(15,19)(17,27)(20,35)(21,30)(22,33)(23,31)(25,32)(26,34)(29,38)(36,40)(37,39);(2,5)(3,8)(6,7)(9,11,15,23)(10,16,19,31)(12,20,26,37)(13,22,14,21)(17,29,32,40)(18,33,28,30)(24,39,34,35)(25,38,27,36);(1,2,4,7)(3,6,8,5)(9,12)(10,17)(11,20)(13,27)(14,25)(15,26)(16,29)(18,34)(19,32)(21,38)(22,36)(23,37)(24,28)(30,39)(31,40)(33,35);(9,13,15,14)(10,18,19,28)(11,21,23,22)(12,25,26,27)(16,30,31,33)(17,24,32,34)(20,36,37,38)(29,35,40,39)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^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, 143772);