#331399

order	512
nilpotency class	3
isoclinism	isoclinic class of #331114
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

709 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=h,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fh,g^c=gi

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, 238595);