#295145

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

689 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202341);