#9063307

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

4437 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8970503);