#426947

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

796 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)(9,10,14,20,16,22,15,21)(11,19,13,29,25,32,24,17)(12,18,27,34,28,36,30,35)(23,33,26,39,38,40,37,31);(1,3,5,4)(2,6,7,8)(9,11,16,25)(10,17,22,29)(12,26,28,37)(13,14,24,15)(18,33,36,40)(19,20,32,21)(23,30,38,27)(31,35,39,34);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,23)(13,26)(14,27)(15,30)(16,28)(17,31)(19,33)(20,34)(21,35)(22,36)(24,37)(25,38)(29,39)(32,40);(9,13,14,25,16,24,15,11)(10,19,20,29,22,32,21,17)(12,26,27,38,28,37,30,23)(18,33,34,39,36,40,35,31)

Polycyclic presentation:

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

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, 334143);