#466709

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fi,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,g^c=gi,h^d=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, 373905);