#9607594

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

4215 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=i,e^2=g,f^2=hi,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, 9514790);