#191255

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

518 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)(3,9,4,8,14,15,13,16)(17,18,22,28,23,27,21,26)(19,29,31,41,32,24,36,39)(20,33,34,40,35,42,37,25)(30,44,46,43,47,48,45,38);(1,3,6,13,7,14,5,4)(2,8,11,15,12,16,10,9)(17,19,21,31,23,32,22,36)(18,24,26,39,27,29,28,41)(20,30,34,45,35,47,37,46)(25,38,33,48,40,43,42,44);(1,4,5,14,7,13,6,3)(2,9,10,16,12,15,11,8)(17,20)(18,25,27,40)(19,30)(21,34)(22,37)(23,35)(24,38,29,43)(26,33,28,42)(31,45)(32,47)(36,46)(39,48,41,44);(17,21,23,22)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(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,g^b=ghi,g^c=ghi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98451);