#116251

order	512
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=hi,g^2=h,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fi,d^c=dh,e^c=ehi,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23447);