#8900120

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

4442 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)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,23)(21,22);(3,7)(5,8)(9,11,15,12)(10,16,14,17)(13,20,22,19)(18,24,21,23)(25,26,30,31)(27,32,36,39)(28,29,33,38)(34,40,37,35);(1,2)(3,5)(4,6)(7,8)(10,14)(16,17)(18,21)(23,24)(26,31)(29,38)(32,39)(35,40);(9,12,15,11)(10,17,14,16)(13,19,22,20)(18,23,21,24)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,34)(38,40);(1,3)(2,5)(4,7)(6,8)(9,13,15,22)(10,18,14,21)(11,19,12,20)(16,23,17,24)(25,28,30,33)(26,29,31,38)(27,34,36,37)(32,40,39,35);(1,2)(3,5)(4,6)(7,8)(9,14,15,10)(11,17,12,16)(13,21,22,18)(19,24,20,23)(25,29)(26,33)(27,35)(28,31)(30,38)(32,34)(36,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8807316);