#500701

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321221, #72321812, #72321850
maximal normal subgroup of	#72321221, #72321812, #72321850

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407897);