#112186

order	512
nilpotency class	3
isoclinism	isoclinic class of #112177
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

793 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,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, 19382);