#265126

order	512
nilpotency class	3
isoclinism	isoclinic class of #265099
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

806 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:

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

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,d^b=dh,f^b=fhi,g^b=ghi,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, 172322);