#9105278

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

4486 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=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, 9012474);