#191245

order	512
nilpotency class	3
isoclinism	isoclinic class of #191243
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

526 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)(3,12)(4,8)(5,10)(6,15)(7,11)(9,14)(13,16)(17,18)(19,29,32,24)(20,33,37,42)(21,26)(22,28)(23,27)(25,36,43,35)(30,45)(31,44,38,40)(34,47)(39,46)(41,48);(1,3,6,4,7,14,5,13)(2,8,10,12,11,16,15,9)(17,19,22,31,23,32,21,38)(18,24,28,40,27,29,26,44)(20,34,36,48,37,30,35,46)(25,41,33,45,43,39,42,47);(1,4,6,14,7,13,5,3)(2,9,10,8,11,12,15,16)(17,20)(18,25,27,43)(19,30)(21,35)(22,36)(23,37)(24,39,29,41)(26,42,28,33)(31,46)(32,34)(38,48)(40,47,44,45);(1,5,7,6)(2,10,11,15)(3,13,14,4)(8,12,16,9)(17,21,23,22)(18,26,27,28)(19,31,32,38)(20,35,37,36)(24,40,29,44)(25,42,43,33)(30,46,34,48)(39,47,41,45)

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,c^b=ch,d^b=dhi,g^b=ghi,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, 98441);