#116797

order	512
nilpotency class	3
isoclinism	isoclinic class of #116795
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

662 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)(9,10,14,18,15,20,13,19)(11,21,24,33,25,32,23,16)(12,26,28,17,30,35,29,36)(22,31,27,40,37,39,38,34);(1,2)(3,7)(4,6)(5,8)(9,11,13,23,15,25,14,24)(10,16,18,21,20,33,19,32)(12,27,28,37,30,38,29,22)(17,34,26,39,36,40,35,31);(2,6)(3,4)(7,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,26)(19,35)(20,36)(21,34)(23,27)(24,38)(25,37)(32,40)(33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23993);