#112242

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 19438);