#503064

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60717	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

739 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410260);