#10171784

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

4470 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:

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

Polycyclic presentation:

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

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, 10078980);