#8743437

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

4754 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8650633);