#9506911

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

4743 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9414107);