#8806857

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

4500 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=ghi,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, 8714053);