#507196

order	512
nilpotency class	4
isoclinism	isoclinic class of #502472
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25441	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

563 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)(3,8)(4,6)(5,7)(9,10)(11,25)(12,26)(13,33)(14,22)(15,21)(16,23)(17,38)(18,24)(19,31)(20,28)(27,37)(29,40)(30,35)(32,34)(36,39);(1,3,5,4)(2,6,7,8)(9,11,14,26,16,19,35,34,18,29,36,13,17,28,15,27)(10,12,30,40,38,37,22,31,24,33,21,25,23,32,39,20);(1,4,5,3)(2,6,7,8)(9,12,14,25,16,37,35,20,18,33,36,40,17,32,15,31)(10,19,21,34,38,29,39,13,24,28,30,27,23,11,22,26);(2,7)(6,8)(9,13,15,29,17,34,36,19,18,26,35,11,16,27,14,28)(10,20,22,37,23,25,30,12,24,31,39,32,38,40,21,33)

Polycyclic presentation:

2^9|b^2=egh,c^2=e,d^2=f,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^b=di,g^c=gi

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, 414392);