#470620

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 377816);