#290900

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

735 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)(9,10,13,16)(11,17,18,24)(12,21,19,15)(14,22,23,20)(26,31)(28,30)(34,40)(35,37);(1,3,4,8)(2,5,6,7)(11,18)(12,19)(15,21)(17,24)(25,26,29,31)(27,34,36,40)(28,33,30,38)(32,37,39,35);(9,11,19,22,13,18,12,20)(10,14,21,24,16,23,15,17)(25,27)(26,32,31,39)(28,37)(29,36)(30,35)(33,34,38,40);(2,6)(5,7)(25,28,29,30)(26,33,31,38)(27,35,36,37)(32,40,39,34)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=hi,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, 198096);