#253663

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
prime radical	#25743	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

892 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)(3,6)(4,7)(5,8)(9,10,14,19)(11,15,21,17)(12,16,24,32)(13,18,27,36)(20,31,37,28)(22,29,38,34)(23,40,39,30)(25,33,26,35);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,23,24,39)(13,22)(14,21)(16,30,32,40)(17,19)(18,29)(20,25,37,26)(27,38)(28,33,31,35)(34,36);(2,6)(3,4)(7,8)(9,12)(10,16)(11,20)(13,26)(14,24)(15,28)(17,31)(18,35)(19,32)(21,37)(22,39)(23,38)(25,27)(29,40)(30,34)(33,36);(9,11)(10,17)(12,20)(13,22)(14,21)(15,19)(16,31)(18,34)(23,25)(24,37)(26,39)(27,38)(28,32)(29,36)(30,35)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,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, 160859);