#8346032

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4368 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:

(1,2,4,6)(3,7,8,5)(10,11)(15,17)(16,18)(22,23)(25,26,32,36)(27,35,30,31)(28,34,29,33)(37,40,38,39);(1,3,4,8)(2,5,6,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,32,30)(26,31,36,35)(28,37,29,38)(33,40,34,39);(1,3)(2,5)(4,8)(6,7)(13,21)(16,18)(19,24)(22,23)(25,28,32,29)(26,33,36,34)(27,37,30,38)(31,40,35,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,29,32,28)(26,34,36,33)(27,37,30,38)(31,40,35,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,30,32,27)(26,35,36,31)(28,38,29,37)(33,39,34,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(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=g,c^2=hi,d^2=gi,e^2=gi,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, 8253228);