#479811

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

629 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,f^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,h^b=hi,d^c=de,e^c=ei,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, 387007);