#8260767

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

4430 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)(3,5,7,8)(11,12)(15,16)(19,20)(22,23)(25,26,32,36)(27,34,28,33)(29,35,30,31)(37,40,38,39);(3,7)(5,8)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,27,32,28)(26,33,36,34)(29,38,30,37)(31,39,35,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,15,18,16)(13,20,21,19)(17,23,24,22)(25,28,32,27)(26,34,36,33)(29,38,30,37)(31,39,35,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,29,32,30)(26,31,36,35)(27,37,28,38)(33,40,34,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22)(25,30,32,29)(26,35,36,31)(27,38,28,37)(33,39,34,40);(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,31,32,35)(26,29,36,30)(27,39,28,40)(33,38,34,37)

Polycyclic presentation:

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

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, 8167963);