#252576

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159772);