#191223

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 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)(3,9)(4,8)(5,10)(6,11)(7,12)(13,16)(14,15)(17,18)(19,29,33,24)(20,34,37,41)(21,26)(22,28)(23,27)(25,36,42,35)(30,44)(31,40,32,39)(38,46)(43,45)(47,48);(1,3,6,13,7,14,5,4)(2,8,10,15,12,16,11,9)(17,19,22,32,23,33,21,31)(18,24,28,40,27,29,26,39)(20,30,36,46,37,47,35,45)(25,38,34,48,42,43,41,44);(1,4,5,14,7,13,6,3)(2,9,11,16,12,15,10,8)(17,20)(18,25,27,42)(19,30)(21,35)(22,36)(23,37)(24,38,29,43)(26,41,28,34)(31,45)(32,46)(33,47)(39,44,40,48);(1,5,7,6)(2,10,12,11)(3,4,14,13)(8,15,16,9)(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)(38,44,43,48)

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,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, 98419);