#8258045

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

4550 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:

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

Polycyclic presentation:

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

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, 8165241);