#253724

order	512
nilpotency class	3
isoclinism	isoclinic class of #253699
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	C₂³×C₄	C₂⁴
prime radical	#3496	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

889 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)(9,10,14,18)(11,15,21,28)(12,16,24,30)(13,17,25,31)(19,26,22,34)(20,27,33,39)(23,36,37,29)(32,40,35,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,22,24,19)(13,20)(14,21)(16,26,30,34)(17,27)(18,28)(23,35,37,32)(25,33)(29,38,36,40)(31,39);(2,6)(3,4)(7,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,26)(17,29)(18,30)(20,32)(21,22)(25,37)(27,38)(28,34)(31,36)(33,35)(39,40);(9,13,14,25)(10,17,18,31)(11,20,21,33)(12,23,24,37)(15,27,28,39)(16,29,30,36)(19,32,22,35)(26,38,34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,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, 160920);