#257008

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59217	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

825 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19)(25,26)(27,35)(28,34)(29,33)(30,32)(31,38)(36,40)(37,39);(1,3,5,4)(2,6,7,8)(10,14)(13,20)(17,24)(19,22)(25,27,29,38)(26,31,33,35)(28,37,30,36)(32,39,34,40);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,18)(13,22)(14,23)(16,24)(17,21)(19,20)(26,32)(27,36)(28,30)(33,34)(37,38)(39,40);(2,7)(6,8)(10,14)(11,18)(16,21)(19,22)(25,28,29,30)(26,32,33,34)(27,37,38,36)(31,39,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 164204);