#9065520

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

4473 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8972716);