#410466

order	512
nilpotency class	3
isoclinism	isoclinic class of #393988
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

881 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)(11,21,23,30,24,34,28,16)(12,17,25,31,27,33,26,32)(22,35,36,39,37,40,38,29);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,22)(13,28)(14,23)(15,24)(17,29)(18,34)(19,21)(20,30)(25,38)(26,36)(27,37)(31,40)(32,35)(33,39);(9,11)(10,16,20,30)(12,22)(13,28)(14,23)(15,24)(17,29,33,39)(18,34,19,21)(25,38)(26,36)(27,37)(31,40,32,35);(2,6)(5,8)(9,12,14,26,15,27,13,25)(10,17,19,32,20,33,18,31)(11,22,23,36,24,37,28,38)(16,29,21,35,30,39,34,40)

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,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317662);