#330182

order	512
nilpotency class	3
isoclinism	isoclinic class of #329395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

680 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)(9,10,13,16,12,14,20,23)(11,17,19,22,18,24,21,15)(25,26,28,32,29,33,37,31)(27,30,34,38,36,40,35,39);(2,5)(6,8)(10,14)(15,22)(16,23)(17,24)(25,27,29,36)(26,30,33,40)(28,34,37,35)(31,39,32,38);(3,7)(6,8)(9,11)(10,15)(12,18)(13,21)(14,22)(16,24)(17,23)(19,20)(26,31)(27,34)(28,37)(32,33)(35,36)(38,39);(1,3,4,7)(2,6,5,8)(11,18)(15,22)(17,24)(19,21)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237378);