#9160757

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

4561 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067953);