#265058

order	512
nilpotency class	3
isoclinism	isoclinic class of #265019
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

798 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)(19,31,21,38)(20,35,37,44)(22,29,23,28)(25,34,27,33)(26,40,42,36)(32,43,39,41)(45,47,48,46);(1,2,6,10)(3,12,13,15)(4,9,5,8)(7,11,16,14)(17,19,23,34,24,21,22,33)(18,25,29,38,30,27,28,31)(20,32,40,47,37,39,36,46)(26,41,44,48,42,43,35,45);(1,3,6,13)(2,7,10,16)(4,14,5,11)(8,12,9,15)(17,20,24,37)(18,26,30,42)(19,32,21,39)(22,36,23,40)(25,41,27,43)(28,35,29,44)(31,45,38,48)(33,46,34,47);(3,13)(7,16)(11,14)(12,15)(17,21,23,33,24,19,22,34)(18,27,29,31,30,25,28,38)(20,32,40,47,37,39,36,46)(26,41,44,48,42,43,35,45)

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,d^c=di,f^c=fi,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, 172254);