#9065681

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

4400 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)(11,19)(12,17)(13,20)(14,15)(16,22)(18,24)(21,23)(25,26,32,35)(27,28,37,29)(30,33,40,38)(31,39,36,34);(1,2)(3,4)(5,6)(7,8)(10,15)(12,23)(16,19)(18,20)(25,27)(26,28)(29,35)(30,31)(32,37)(33,39)(34,38)(36,40);(2,6)(3,7)(10,15)(12,23)(16,19)(18,20)(25,28,32,29)(26,27,35,37)(30,39,40,34)(31,38,36,33);(3,7)(4,8)(9,11)(10,16)(12,18)(13,17)(14,22)(15,19)(20,23)(21,24)(25,29,32,28)(26,27,35,37)(30,34,40,39)(31,38,36,33);(1,3)(2,4)(5,7)(6,8)(9,12,14,23)(10,17,15,21)(11,20,22,18)(13,16,24,19)(25,30)(26,33)(27,36)(28,34)(29,39)(31,37)(32,40)(35,38);(1,4)(2,3)(5,8)(6,7)(9,13,14,24)(10,18,15,20)(11,21,22,17)(12,16,23,19)(25,31,32,36)(26,34,35,39)(27,30,37,40)(28,38,29,33)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=h,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, 8972877);