#191276

order	512
nilpotency class	3
isoclinism	isoclinic class of #191275
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

525 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=ghi,d^c=dh,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98472);