#8127033

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

4628 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=gh,e^2=gh,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, 8034229);