#8806913

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

4565 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714109);