#8761934

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

4721 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8669130);