#291380

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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

731 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)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(25,26)(27,30)(28,31)(29,37)(32,36)(33,38)(34,40)(35,39);(1,3,4,8)(2,5,6,7)(9,11,15,20)(10,16,17,24)(12,21,22,18)(13,19,14,23)(25,27)(26,30)(28,35)(29,33)(31,39)(32,40)(34,36)(37,38);(9,12,13,10,15,22,14,17)(11,18,23,16,20,21,19,24)(26,31)(27,33,35,34)(30,38,39,40)(32,37);(2,6)(5,7)(9,13,15,14)(10,12,17,22)(11,19,20,23)(16,21,24,18)

Polycyclic presentation:

2^9|b^2=eh,c^2=fi,d^2=h,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=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198576);