#258432

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	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

837 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19)(25,26,30,34)(27,31,37,40)(28,32,29,33)(35,38,36,39);(2,5)(6,7)(10,14)(15,23)(16,21)(17,24)(25,27)(26,31)(28,36)(29,35)(30,37)(32,39)(33,38)(34,40);(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)(26,32)(27,35)(28,29)(33,34)(36,37)(38,39);(1,3,4,8)(2,6,5,7)(10,14)(15,23)(16,21)(17,24)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,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, 165628);