#504595

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

495 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)(11,19)(12,24)(13,16)(14,18)(15,21)(17,26)(20,30)(22,31)(23,36)(25,34)(27,33)(28,32)(29,38)(35,40)(37,39);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,28)(16,19)(17,30)(18,36)(22,25)(23,27)(24,29)(26,38)(31,35)(32,33)(34,37)(39,40);(1,3,4,7)(2,6,5,8)(10,16)(11,20)(12,25)(14,27)(15,29)(17,34)(19,30)(21,38)(22,23)(24,35)(26,40)(28,39)(31,36)(32,37);(2,5)(6,8)(9,12,14,25,13,26,27,40)(10,17,18,35,16,24,33,34)(11,20,28,22,21,38,23,39)(15,30,36,31,19,29,32,37)

Polycyclic presentation:

2^9|c^2=ei,d^2=fhi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=gh,h^c=hi,g^d=gi

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, 411791);