#319203

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egh,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gi

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, 226399);