#8658458

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

4384 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,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, 8565654);