#8936685

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

4507 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=h,e^2=g,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, 8843881);