#291015

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

770 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)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35);(1,3,4,8)(2,5,6,7)(11,18)(12,21)(15,20)(17,23)(25,27)(26,31)(28,35)(29,37)(30,34)(32,33)(36,39)(38,40);(9,11,12,19,13,18,21,24)(10,14,15,17,16,22,20,23)(25,28)(26,32)(27,33)(29,38)(30,36)(31,39)(34,40)(35,37);(2,6)(5,7)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=i,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,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198211);