#119935

order	512
nilpotency class	3
isoclinism	isoclinic class of #119934
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	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

524 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)(4,8)(5,15)(6,10)(7,12)(9,14)(13,16)(17,18)(19,28)(20,32,36,46)(21,37)(22,26)(23,27)(24,30)(25,35,43,34)(29,39,33,42)(31,40)(38,41)(44,45,48,47);(1,3,6,13)(2,7,10,8)(4,14,12,5)(9,16,15,11)(17,19)(18,24)(20,33)(21,38)(22,30)(23,31)(25,42)(26,40)(27,41)(28,37)(29,36)(32,44)(34,45)(35,47)(39,43)(46,48);(1,4,5,3,6,12,14,13)(2,8,9,16,10,7,15,11)(17,20,21,34,23,36,22,35)(18,25,26,32,27,43,37,46)(19,29,30,45,31,33,38,47)(24,39,28,44,41,42,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27131);