#8850655

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

4799 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8757851);