#470621

order	512
nilpotency class	3
isoclinism	isoclinic class of #460330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

838 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)(3,6)(4,5)(7,8)(9,10,16,19,15,20,14,31)(11,21,25,34,26,39,24,17)(12,18,13,30,29,35,28,36)(22,33,27,38,23,37,40,32);(2,5)(6,8)(9,11,14,24,15,26,16,25)(10,17,19,21,20,34,31,39)(12,27,28,22,29,40,13,23)(18,33,30,38,35,37,36,32);(3,7)(6,8)(9,12,15,29)(10,18,20,35)(11,22,26,23)(13,16,28,14)(17,32,34,38)(19,36,31,30)(21,37,39,33)(24,27,25,40);(1,3,4,7)(2,6,5,8)(9,13,14,12,15,28,16,29)(10,18,19,30,20,35,31,36)(11,23,24,27,26,22,25,40)(17,33,21,38,34,37,39,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,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,f^a=fi,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=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, 377817);