#235185

order	512
nilpotency class	3
isoclinism	isoclinic class of #235159
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	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1362 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=hi,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,e^c=ei,f^c=fh

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, 142381);