#9210349

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

4278 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=i,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9117545);