#507194

order	512
nilpotency class	4
isoclinism	isoclinic class of #502469
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

565 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)(11,26)(12,32)(13,18)(14,27)(15,39)(16,23)(17,25)(19,20)(21,37)(22,38)(24,28)(29,34)(30,40)(31,33)(35,36);(1,3,5,4)(2,6,7,8)(9,11,14,13,28,19,15,29,17,31,38,32,16,30,37,36)(10,18,39,33,23,35,27,20,25,12,21,26,24,34,22,40);(1,4,5,3)(2,6,7,8)(9,12,14,33,28,34,15,20,17,18,38,26,16,35,37,40)(10,19,21,29,23,31,22,32,25,30,39,36,24,11,27,13);(2,7)(6,8)(9,13,15,31,16,36,14,19,17,32,37,11,28,29,38,30)(10,20,22,35,24,33,21,18,25,40,27,34,23,26,39,12)

Polycyclic presentation:

2^9|b^2=eghi,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,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414390);