#8506023

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413219);