#8963683

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

4425 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)(3,5)(4,8)(6,7)(9,10,13,16)(11,14,17,18)(12,15,20,22)(19,21,23,24);(2,7)(3,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,26,30,28)(27,31,34,35)(29,32,37,36)(33,38,39,40);(1,3,6,8)(2,4,7,5)(11,17)(12,20)(14,18)(15,22)(26,28)(29,37)(31,35)(33,39);(1,4,6,5)(2,3,7,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,27,30,34)(26,31,28,35)(29,33,37,39)(32,38,36,40);(1,2)(3,4)(5,8)(6,7)(9,11,13,17)(10,14,16,18)(12,19,20,23)(15,21,22,24)(25,28,30,26)(27,31,34,35)(29,36,37,32)(33,38,39,40);(1,5,6,4)(2,8,7,3)(9,12,13,20)(10,15,16,22)(11,19,17,23)(14,21,18,24)(25,29)(26,32)(27,33)(28,36)(30,37)(31,38)(34,39)(35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8870879);