#265010

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

785 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)(3,11)(4,8)(5,9)(6,10)(7,14)(12,15)(13,16)(17,18,24,30)(19,31,34,27)(20,35,38,44)(21,25,33,41)(22,29,23,28)(26,39,43,37)(32,46,47,40)(36,45,48,42);(1,2)(3,12)(4,8)(5,9)(6,10)(7,13)(11,15)(14,16)(17,19,23,33,24,34,22,21)(18,25,29,27,30,41,28,31)(20,32,39,48,38,47,37,36)(26,40,44,42,43,46,35,45);(1,3,6,15)(2,7,10,16)(4,13,5,14)(8,11,9,12)(17,20,24,38)(18,26,30,43)(19,32,34,47)(21,36,33,48)(22,37,23,39)(25,40,41,46)(27,42,31,45)(28,35,29,44);(1,4,6,5)(2,8,10,9)(3,13,15,14)(7,11,16,12)(17,21,22,34,24,33,23,19)(18,27,28,25,30,31,29,41)(20,36,37,47,38,48,39,32)(26,42,35,40,43,45,44,46)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=ehi,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, 172206);