#191268

order	512
nilpotency class	3
isoclinism	isoclinic class of #191234
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	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

491 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98464);