#8733433

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

4462 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8640629);