#9210528

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

4544 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9117724);