#253694

order	512
nilpotency class	3
isoclinism	isoclinic class of #253651
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=ghi,f^d=fh

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, 160890);