#238212

order	512
nilpotency class	3
isoclinism	isoclinic class of #234006
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

858 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,d^c=dh,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, 145408);