#397501

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1305 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=i,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 304697);