#8127281

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

4289 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)(11,18)(12,15)(13,21)(14,17)(16,22)(19,20)(23,24)(25,26)(27,28)(29,32)(30,37)(31,33)(34,39)(35,40)(36,38);(3,7)(6,8)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(25,27,32,31)(26,33,29,28)(30,35,39,38)(34,40,37,36);(1,2)(3,6)(4,5)(7,8)(9,12,14,18)(10,15,17,11)(13,20,22,24)(16,23,21,19)(25,28)(26,31)(27,29)(30,36)(32,33)(34,38)(35,37)(39,40);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,20)(14,18)(15,17)(16,19)(21,23)(22,24)(25,29,32,26)(27,33,31,28)(30,37,39,34)(35,40,38,36);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,22)(23,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40);(9,13,14,22)(10,16,17,21)(11,19,15,23)(12,20,18,24)(25,31,32,27)(26,33,29,28)(30,38,39,35)(34,40,37,36)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^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, 8034477);