#413067

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1522 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 320263);