#475716

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

721 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,26)(13,16)(14,19)(15,20)(17,24)(18,30)(22,35)(23,31)(25,32)(27,34)(28,33)(29,36)(37,39)(38,40);(1,3,5,4)(2,6,7,8)(10,16,20,19)(12,27)(17,32,21,31)(18,34,36,26)(22,38)(28,39)(29,30)(33,35,37,40);(3,4)(6,8)(9,11)(10,17,20,21)(12,28)(13,25)(14,23)(15,24)(16,31,19,32)(18,35,36,40)(22,29)(26,37,34,33)(27,39)(30,38);(1,4,5,3)(2,6,7,8)(9,12,13,29,15,27,14,30)(10,18,19,34,20,36,16,26)(11,22,23,28,24,38,25,39)(17,33,31,35,21,37,32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382912);