#8968131

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

4448 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)(3,4)(5,6)(7,8)(9,10)(11,12)(13,21)(14,15)(16,19)(17,22)(18,23)(20,24)(25,26,31,29)(27,32,34,28)(30,37,38,33)(35,36,40,39);(2,6)(4,7)(10,15)(12,16)(17,21)(20,23)(25,27,31,34)(26,32,29,28)(30,35,38,40)(33,39,37,36);(1,2,5,6)(3,7,8,4)(10,15)(12,16)(13,22)(18,24)(26,29)(27,34)(30,38)(36,39);(1,3)(2,7)(4,6)(5,8)(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,23,21,20)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,40)(38,39);(1,4,5,7)(2,3,6,8)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,29,31,26)(27,32,34,28)(30,37,38,33)(35,39,40,36);(1,3)(2,7)(4,6)(5,8)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=hi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8875327);