#112244

order	512
nilpotency class	3
isoclinism	isoclinic class of #112221
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

785 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)(3,10)(4,6,13,8)(7,14)(11,16)(12,15)(17,18,22,28,23,26,21,27)(19,29,32,40,33,24,31,41)(20,34,37,25,38,44,36,43)(30,42,46,45,47,48,35,39);(2,6)(3,11)(4,13)(8,9)(10,15)(12,14)(17,19,23,33)(18,24,26,29)(20,35,38,46)(21,32,22,31)(25,42,43,48)(27,41,28,40)(30,37,47,36)(34,45,44,39);(1,3,5,12)(2,7,9,16)(4,14,13,11)(6,15,8,10)(17,20,23,38)(18,25,26,43)(19,30,33,47)(21,37,22,36)(24,39,29,45)(27,44,28,34)(31,46,32,35)(40,42,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=h,f^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,f^b=fh,g^b=ghi,d^c=di,e^c=eh,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, 19440);