#9508748

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

4417 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(26,31)(27,35)(33,40)(34,37);(1,3,5,7)(2,6,4,8)(11,17)(15,21)(18,23)(22,24)(25,26,30,31)(27,32,35,39)(28,33,36,40)(29,34,38,37);(2,4)(3,7)(10,14)(15,21)(16,19)(22,24)(28,36)(29,38)(33,40)(34,37);(1,3,5,7)(2,6,4,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,32)(28,37)(29,33)(30,35)(31,39)(34,36)(38,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,30,36)(26,33,31,40)(27,34,35,37)(29,32,38,39);(1,4,5,2)(3,8,7,6)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29,30,38)(26,34,31,37)(27,33,35,40)(28,32,36,39)

Polycyclic presentation:

2^9|b^2=gi,d^2=i,e^2=g,f^2=gh,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, 9415944);