#409831

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

820 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=f,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,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, 317027);