#323196

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

624 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,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, 230392);