#8634846

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

4475 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8542042);