#112031

order	512
nilpotency class	3
isoclinism	isoclinic class of #112011
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
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

1161 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=di,e^c=eh,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, 19227);