#322349

order	512
nilpotency class	3
isoclinism	isoclinic class of #322325
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

612 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:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229545);