#264978

order	512
nilpotency class	3
isoclinism	isoclinic class of #264939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#546
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172174);