#330773

order	512
nilpotency class	3
isoclinism	isoclinic class of #329455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

581 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,g^b=gi,d^c=de,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, 237969);