#119173

order	512
nilpotency class	3
isoclinism	isoclinic class of #119168
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

536 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=i,e^2=i,f^2=h,g^2=i,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=ehi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26369);