#8599366

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
prime radical	#37825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4469 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)(11,16)(12,13)(14,19)(15,17)(18,23)(20,24)(21,22)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(2,5)(6,8)(9,11,15,20)(10,16,17,24)(12,21,18,19)(13,22,23,14)(26,30)(31,33)(32,37)(38,39);(1,2,4,5)(3,6,7,8)(10,17)(11,20)(13,23)(19,21)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(3,7)(6,8)(9,12,15,18)(10,13,17,23)(11,19,20,21)(14,24,22,16)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,14)(12,17)(15,23)(16,21)(19,24)(20,22)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,3)(2,6)(4,7)(5,8)(9,14)(10,19)(11,13)(12,16)(15,22)(17,21)(18,24)(20,23)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=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, 8506562);