#411237

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

735 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=eg,b^2=gh,c^2=g,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=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, 318433);