#9028718

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
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8935914);