#191232

order	512
nilpotency class	3
isoclinism	isoclinic class of #191231
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:

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

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,d^b=dh,g^b=ghi,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, 98428);