#8417266

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

4249 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324462);