#323148

order	512
nilpotency class	3
isoclinism	isoclinic class of #323133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#550
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

606 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=hi,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=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, 230344);