#8891548

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

4271 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,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, 8798744);