#121170

order	512
nilpotency class	3
isoclinism	isoclinic class of #121169
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

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, 28366);