#8657123

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

4434 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)(3,7,8,5)(9,10,15,12)(11,16,13,17)(14,18,22,20)(19,23,21,24)(25,26)(27,33)(28,38)(29,30)(31,34)(32,35)(36,40)(37,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,17)(13,15)(14,19)(18,23)(20,24)(21,22)(25,27,32,37)(26,33,35,39)(28,34,29,36)(30,40,38,31);(1,3)(2,5)(4,8)(6,7)(9,10,15,12)(11,17,13,16)(14,18,22,20)(19,24,21,23)(25,28,32,29)(26,30,35,38)(27,36,37,34)(31,39,40,33);(1,3)(2,5)(4,8)(6,7)(9,12,15,10)(11,17,13,16)(14,18,22,20)(19,23,21,24)(25,29,32,28)(26,30,35,38)(27,34,37,36)(31,39,40,33);(1,2)(3,5)(4,6)(7,8)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,30)(26,28)(27,31)(29,35)(32,38)(33,36)(34,39)(37,40);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,31)(26,34)(27,30)(28,39)(29,33)(32,40)(35,36)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8564319);