#8811193

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

4266 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)(3,4)(5,6)(7,8)(10,13)(15,18)(16,21)(22,24)(25,26,31,34)(27,33,29,32)(28,37,39,30)(35,36,38,40);(2,6)(4,7)(12,20)(16,21)(17,23)(22,24)(25,27,31,29)(26,32,34,33)(28,38,39,35)(30,40,37,36);(2,6)(3,8)(9,10,14,13)(11,15,19,18)(12,16,20,21)(17,22,23,24)(25,28)(26,30)(27,35)(29,38)(31,39)(32,36)(33,40)(34,37);(1,3,5,8)(2,7,6,4)(9,11,14,19)(10,15,13,18)(12,17,20,23)(16,22,21,24)(25,29,31,27)(26,33,34,32)(28,38,39,35)(30,40,37,36);(1,2)(3,7)(4,8)(5,6)(9,12,14,20)(10,16,13,21)(11,17,19,23)(15,22,18,24)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39);(1,4,5,7)(2,8,6,3)(9,13,14,10)(11,18,19,15)(12,21,20,16)(17,24,23,22)(25,26)(27,32)(28,30)(29,33)(31,34)(35,36)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=h,e^2=gh,f^2=hi,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, 8718389);