#10047511

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

4730 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gi,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, 9954707);