#112120

order	512
nilpotency class	3
isoclinism	isoclinic class of #112109
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

787 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=di,f^c=fh,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, 19316);