#8322390

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 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)(11,16)(12,14)(13,17)(15,19)(18,23)(20,21)(22,24)(25,26,32,36)(27,35,30,31)(28,34,29,33)(37,40,38,39);(2,5)(6,7)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,27,32,30)(26,31,36,35)(28,37,29,38)(33,40,34,39);(1,2)(3,6)(4,5)(7,8)(13,21)(17,20)(18,24)(22,23)(25,28,32,29)(26,33,36,34)(27,37,30,38)(31,40,35,39);(1,3,4,8)(2,6,5,7)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,29,32,28)(26,34,36,33)(27,37,30,38)(31,40,35,39);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,30,32,27)(26,35,36,31)(28,38,29,37)(33,39,34,40);(9,13,14,21)(10,17,12,20)(11,18,19,24)(15,22,16,23)(25,31,32,35)(26,27,36,30)(28,39,29,40)(33,38,34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8229586);