#320300

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
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

745 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)(3,7)(4,5)(6,8)(9,10,12,16,13,15,21,23)(11,17,19,22,18,24,20,14)(25,26,30,32,29,33,28,34)(27,31,35,40,37,38,36,39);(2,5)(6,7)(11,18)(14,22)(17,24)(19,20)(25,27,28,36,29,37,30,35)(26,31,34,39,33,38,32,40);(9,11,13,18)(10,14,15,22)(12,20,21,19)(16,24,23,17)(26,32)(27,35)(28,30)(31,38)(33,34)(36,37);(1,3,4,8)(2,6,5,7)(10,15)(14,22)(16,23)(17,24)(26,33)(31,38)(32,34)(39,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=ghi,c^2=hi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,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, 227496);