#465258

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

661 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372454);