#410467

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=g,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317663);