#328786

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

788 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)(9,10,13,17)(11,18,14,15)(12,20,21,24)(16,22,23,19)(25,26)(27,35)(28,33)(29,31)(30,34)(32,37)(36,39)(38,40);(2,5)(6,8)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,20,23,24)(26,31)(27,36)(28,30)(32,35)(37,38)(39,40);(3,7)(6,8)(9,12,13,21)(10,16,17,23)(11,19,14,22)(15,20,18,24)(25,27,29,38)(26,32,31,39)(28,36,30,37)(33,35,34,40);(1,3,4,7)(2,6,5,8)(12,21)(16,23)(19,22)(20,24)(25,28,29,30)(26,33,31,34)(27,37,38,36)(32,40,39,35)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235982);