#8371276

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

4424 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8278472);