#10047401

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

4520 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)(11,14)(12,15)(13,18)(16,22)(17,19)(20,24)(21,23)(25,26,31,29)(27,35,37,32)(28,33,38,39)(30,36,40,34);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,17)(13,16)(15,19)(18,22)(20,24)(21,23)(25,27,31,37)(26,32,29,35)(28,34,38,36)(30,33,40,39);(1,3,6,5)(2,8,4,7)(10,11)(12,19)(16,18)(20,21)(26,29)(28,38)(32,35)(34,36);(1,2,6,4)(3,8,5,7)(9,11,14,10)(12,17,19,15)(13,18,22,16)(20,24,21,23)(25,28,31,38)(26,33,29,39)(27,36,37,34)(30,35,40,32);(1,4,6,2)(3,8,5,7)(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,29,31,26)(27,32,37,35)(28,39,38,33)(30,34,40,36);(1,5,6,3)(2,7,4,8)(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,30)(26,34)(27,33)(28,35)(29,36)(31,40)(32,38)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9954597);