#8550912

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

4367 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8458108);