#326545

order	512
nilpotency class	3
isoclinism	isoclinic class of #326535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2563
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

461 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=di,f^c=fi,g^c=gh

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, 233741);