#112032

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

867 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,10,6,7,4,9)(3,11,13,8,14,16,12,15)(17,18)(19,28)(20,33,36,42)(21,37)(22,26)(23,27)(24,30)(25,35,40,34)(29,38,46,48)(31,39)(32,41)(43,45,47,44);(2,7)(4,5)(11,16)(12,13)(17,19,23,32)(18,24,27,39)(20,29,36,46)(21,31,22,30)(25,38,40,48)(26,41,37,28)(33,43,42,47)(34,45,35,44);(1,3,6,14)(2,8,7,15)(4,13,5,12)(9,16,10,11)(17,20,23,36)(18,25,27,40)(19,29,32,46)(21,35,22,34)(24,38,39,48)(26,42,37,33)(28,43,41,47)(30,45,31,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,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, 19228);