#8726401

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

4393 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)(9,10,15,13)(11,16,18,12)(14,17,23,21)(19,22,24,20)(25,26,31,32)(27,30,33,40)(28,29,37,38)(34,39,36,35);(2,3)(6,7)(9,11,15,18)(10,16,13,12)(14,19,23,24)(17,22,21,20)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,37)(38,39);(1,3,5,2)(4,7,8,6)(11,18)(12,16)(19,24)(20,22)(26,32)(27,33)(28,37)(35,39);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,18)(14,22)(15,16)(17,24)(19,21)(20,23)(25,28)(26,29)(27,34)(30,39)(31,37)(32,38)(33,36)(35,40);(9,13,15,10)(11,12,18,16)(14,17,23,21)(19,22,24,20)(25,29)(26,28)(27,35)(30,36)(31,38)(32,37)(33,39)(34,40);(9,14,15,23)(10,17,13,21)(11,19,18,24)(12,20,16,22)(25,30,31,40)(26,33,32,27)(28,35,37,39)(29,34,38,36)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633597);