#8746605

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

4340 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653801);