#8156817

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

4719 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)(11,16)(12,21)(13,23)(14,17)(15,18)(19,20)(22,24)(25,26)(27,33)(28,31)(29,32)(30,36)(34,38)(35,39)(37,40);(3,7)(6,8)(11,18)(15,16)(19,22)(20,24)(26,28)(32,33)(34,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,18)(10,15,17,16)(12,19,23,22)(13,20,21,24)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,22)(13,20)(14,18)(15,17)(19,23)(21,24)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,19)(14,23)(15,20)(16,24)(17,21)(18,22)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(9,13)(10,12)(11,20)(14,21)(15,19)(16,22)(17,23)(18,24)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^2=g,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^d=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, 8064013);