#8936673

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

4496 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,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, 8843869);