#8722917

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

4491 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8630113);