#8230661

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

4447 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26,31,35)(27,36,30,32)(28,34,29,33)(37,39,38,40);(1,2,4,6)(3,5,8,7)(10,15)(12,16)(17,22)(20,23)(25,27,31,30)(26,32,35,36)(28,37,29,38)(33,39,34,40);(1,2)(3,5)(4,6)(7,8)(13,21)(17,22)(18,24)(20,23)(25,28,31,29)(26,33,35,34)(27,37,30,38)(32,39,36,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,29,31,28)(26,34,35,33)(27,37,30,38)(32,39,36,40);(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,26)(27,36)(28,33)(29,34)(30,32)(31,35)(37,40)(38,39);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30,31,27)(26,32,35,36)(28,37,29,38)(33,40,34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=hi,e^2=ghi,f^2=h,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, 8137857);