#119167

order	512
nilpotency class	3
isoclinism	isoclinic class of #119158
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
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

558 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)(3,11,14,16)(4,7)(5,9)(6,10)(8,13,15,12)(17,18,22,28,23,26,21,27)(19,29,32,40,33,24,31,41)(20,34,37,44,36,25,38,43)(30,39,35,42,46,45,47,48);(2,7)(3,12)(4,5)(8,15)(9,10)(13,14)(17,19,23,33)(18,24,26,29)(20,35,36,47)(21,32,22,31)(25,42,34,48)(27,41,28,40)(30,38,46,37)(39,44,45,43);(1,3,4,13,6,14,5,12)(2,8,9,16,10,15,7,11)(17,20)(18,25,26,34)(19,30)(21,38)(22,37)(23,36)(24,39,29,45)(27,44,28,43)(31,47)(32,35)(33,46)(40,48,41,42)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=d,d^2=i,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,e^c=ehi,f^c=fhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26363);