#8127444

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

4328 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034640);