#331429

order	512
nilpotency class	3
isoclinism	isoclinic class of #331190
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

741 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)(9,10,14,17,13,16,21,24)(11,12,19,20,18,15,23,22)(25,26)(27,30)(28,31)(29,32)(33,37)(34,40)(35,39)(36,38);(2,5)(6,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,24)(21,23)(25,27,33,34,28,35,29,36)(26,30,37,40,31,39,32,38);(3,7)(6,8)(9,12)(10,11)(13,15)(14,22)(16,18)(17,23)(19,24)(20,21)(26,31)(27,34)(29,33)(30,38)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(11,18)(12,15)(19,23)(20,22)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238625);