#191228

order	512
nilpotency class	3
isoclinism	isoclinic class of #8794
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	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

543 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)(19,29,33,24)(20,34,38,43)(21,26)(22,28)(23,27)(25,37,44,36)(30,45)(31,41,32,40)(35,48)(39,47)(42,46);(1,3,6,4,7,14,5,13)(2,8,10,12,11,16,15,9)(17,19,22,32,23,33,21,31)(18,24,28,41,27,29,26,40)(20,35,37,46,38,30,36,47)(25,42,34,45,44,39,43,48);(1,4,6,14,7,13,5,3)(2,9,10,8,11,12,15,16)(17,20)(18,25,27,44)(19,30)(21,36)(22,37)(23,38)(24,39,29,42)(26,43,28,34)(31,46)(32,47)(33,35)(40,45,41,48);(1,5,7,6)(2,10,11,15)(3,13,14,4)(8,12,16,9)(17,21,23,22)(18,26,27,28)(19,31,33,32)(20,36,38,37)(24,40,29,41)(25,43,44,34)(30,46,35,47)(39,45,42,48)

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,c^b=ch,g^b=ghi,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, 98424);