#10047083

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

4487 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)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,30,31)(27,32,35,38)(28,33,34,29)(36,39,40,37);(2,3)(6,7)(11,17)(15,21)(18,23)(22,24)(26,31)(27,35)(28,34)(37,39);(1,3,5,2)(4,7,8,6)(10,14)(15,21)(16,19)(22,24)(26,31)(27,35)(28,34)(37,39);(1,4,5,8)(2,6,3,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,39)(34,40);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28)(26,33)(27,36)(29,31)(30,34)(32,39)(35,40)(37,38);(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29,30,33)(26,34,31,28)(27,37,35,39)(32,40,38,36)

Polycyclic presentation:

2^9|a^2=g,c^2=i,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, 9954279);