#475827

order	512
nilpotency class	3
isoclinism	isoclinic class of #458125
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	#2537
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

908 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,d^2=h,f^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,d^c=de,f^c=fi,g^c=gi

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, 383023);