#409657

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

765 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)(3,7)(4,6)(5,8)(9,10,15,22,16,21,14,20)(11,23,25,18,26,33,12,17)(13,28,30,36,31,35,29,19)(24,32,38,37,39,34,27,40);(1,3,5,4)(2,6,8,7)(9,11,16,26)(10,17)(12,14,25,15)(13,24,31,39)(18,21)(19,32)(20,23)(22,33)(27,29,38,30)(28,40)(34,36)(35,37);(1,3,5,4)(2,7,8,6)(9,12,14,11,16,25,15,26)(10,18,20,33,21,17,22,23)(13,27,29,24,31,38,30,39)(19,34,35,40,36,32,28,37);(1,4,5,3)(2,6,8,7)(9,13,14,29,16,31,15,30)(10,19,22,28,21,36,20,35)(11,24,25,38,26,39,12,27)(17,32,33,40,18,34,23,37)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi

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, 316853);