#9498539

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

4618 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=i,e^2=g,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, 9405735);