#116173

order	512
nilpotency class	3
isoclinism	isoclinic class of #116168
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
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

585 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)(3,10)(4,14)(5,8)(6,9)(7,12)(11,16)(13,15)(17,18,22,27,23,28,21,26)(19,24,30,39,31,40,37,44)(20,32,36,25,34,43,35,42)(29,45,46,41,33,48,47,38);(1,2)(3,10,11,16)(4,14)(5,8)(6,9)(7,13,15,12)(17,19)(18,24)(20,33,34,29)(21,37)(22,30)(23,31)(25,41,42,38)(26,44)(27,39)(28,40)(32,45,43,48)(35,46,36,47);(1,3,6,11)(2,7,9,15)(4,13,5,12)(8,16,14,10)(17,20)(18,25)(19,29)(21,36)(22,35)(23,34)(24,38)(26,43)(27,32)(28,42)(30,47)(31,33)(37,46)(39,48)(40,41)(44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fhi,d^c=dh,e^c=eh,f^c=fi,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, 23369);