#291670

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

782 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)(11,20)(12,21)(13,19)(14,18)(15,17)(16,22)(23,24)(25,26,30,34)(27,35,37,39)(28,33,29,32)(31,36,40,38);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,17)(13,21)(14,22)(15,24)(18,20)(19,23)(27,36)(28,29)(31,39)(32,33)(35,40)(37,38);(9,12,15,20,14,23,13,16)(10,11,19,21,18,22,17,24)(25,27,29,36,30,37,28,38)(26,31,33,39,34,40,32,35);(2,6)(5,7)(9,13,14,15)(10,17,18,19)(11,21,22,24)(12,20,23,16)(27,37)(31,40)(35,39)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198866);