#350945

order	512
nilpotency class	3
isoclinism	isoclinic class of #347668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

568 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)(3,6)(4,5)(7,8)(9,10,14,16)(11,17,19,24)(12,13,21,18)(15,22,23,20)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,40)(38,39);(2,5)(6,8)(9,11,18,22,14,19,13,20)(10,15,21,24,16,23,12,17)(26,30)(31,39)(32,37)(33,40);(1,3,4,7)(2,6,5,8)(9,12,13,16,14,21,18,10)(11,17,20,23,19,24,22,15);(11,19)(15,23)(17,24)(20,22)(25,27,28,34)(26,31,30,39)(29,38,36,35)(32,40,37,33)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=ei,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gi,d^b=dh,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258141);