#328340

order	512
nilpotency class	3
isoclinism	isoclinic class of #326869
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×#496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

753 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)(9,10,15,11)(12,20,24,21)(13,19,14,17)(16,22,18,23)(25,26)(27,31)(28,34)(29,33)(30,35)(32,36)(37,40)(38,39);(2,5)(6,8)(9,11,15,10)(12,21,24,20)(13,19,14,17)(16,22,18,23)(25,27,30,37)(26,31,35,40)(28,36,29,38)(32,33,39,34);(3,7)(6,8)(9,12)(10,16)(11,18)(13,22)(14,23)(15,24)(17,21)(19,20)(25,27)(26,32)(28,36)(29,38)(30,37)(31,33)(34,40)(35,39);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,17,11,19)(12,22,24,23)(16,21,18,20)(25,28,30,29)(26,33,35,34)(27,36,37,38)(31,39,40,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,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, 235536);