#328769

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
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	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

811 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19)(25,26,29,32)(27,30,34,39)(28,36,37,31)(33,40,35,38);(2,5)(6,8)(10,14)(15,23)(16,21)(17,24)(25,27,29,34)(26,30,32,39)(28,33,37,35)(31,38,36,40);(3,7)(6,8)(9,11)(10,15)(12,18)(13,22)(14,23)(16,24)(17,21)(19,20)(27,33)(28,37)(30,38)(31,36)(34,35)(39,40);(1,3,4,7)(2,6,5,8)(11,18)(15,23)(17,24)(19,22)(27,34)(30,39)(33,35)(38,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=eh,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=de,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235965);