#328918

order	512
nilpotency class	3
isoclinism	isoclinic class of #326835
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

676 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236114);