#479882

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

621 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)(3,6)(4,5)(7,8)(9,10,15,18,13,19,14,29)(11,20,23,32,22,16,24,36)(12,25,27,34,28,17,30,35)(21,37,39,40,38,31,26,33);(2,5)(3,7)(9,11,14,24,13,22,15,23)(10,16,18,36,19,20,29,32)(12,26,27,21,28,39,30,38)(17,33,34,31,25,40,35,37);(3,7)(6,8)(9,12,15,30,13,28,14,27)(10,17,29,35,19,25,18,34)(11,21,24,39,22,38,23,26)(16,31,36,40,20,37,32,33);(1,3,4,7)(2,6,5,8)(9,10)(11,16,22,20)(12,17)(13,19)(14,18)(15,29)(21,37,38,31)(23,32,24,36)(25,28)(26,40,39,33)(27,34)(30,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=gi,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387078);