#290930

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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	#45986	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

771 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)(4,6)(5,8)(9,10,15,11)(12,20,21,24)(13,19,14,17)(16,23,18,22)(25,26,31,35)(27,32,33,39)(28,37,40,36)(29,34,30,38);(1,3,4,8)(2,5,6,7)(9,11,15,10)(12,20,21,24)(13,19,14,17)(16,22,18,23)(25,27,31,33)(26,32,35,39)(28,38,40,34)(29,36,30,37);(9,12,14,23,15,21,13,22)(10,16,19,24,11,18,17,20)(25,28,29,39,31,40,30,32)(26,33,38,36,35,27,34,37);(2,6)(5,7)(12,21)(16,18)(20,24)(22,23)(25,29,31,30)(26,34,35,38)(27,36,33,37)(28,39,40,32)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198126);