#117268

order	512
nilpotency class	3
isoclinism	isoclinic class of #117259
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=hi,d^2=h,e^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^b=eh,f^b=fi,d^c=dh,e^c=eh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24464);