#255500

order	512
nilpotency class	3
isoclinism	isoclinic class of #247047
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1040 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24)(25,26,28,32)(27,34,36,30)(29,31,38,33)(35,40,37,39);(2,5)(6,8)(25,27)(26,30)(28,36)(29,35)(31,39)(32,34)(33,40)(37,38);(1,3,4,7)(2,6,5,8)(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18)(26,31)(27,35)(29,38)(32,33)(36,37)(39,40);(2,5)(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(26,32)(30,34)(31,33)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 162696);