#10040443

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

4344 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=i,f^2=h,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, 9947639);