#8267623

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

4431 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=hi,e^2=i,f^2=i,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, 8174819);