#8924749

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

4467 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)(3,5)(4,7)(6,8)(10,15)(11,17)(16,23)(18,19);(1,2,6,8)(3,5,7,4)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,26,31,34)(27,35,29,32)(28,30,33,40)(36,39,38,37);(1,3,6,7)(2,4,8,5)(11,17)(13,22)(18,19)(20,24)(25,27,31,29)(26,32,34,35)(28,38,33,36)(30,39,40,37);(1,4,6,5)(2,3,8,7)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,37,35,39);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,29,31,27)(26,32,34,35)(28,36,33,38)(30,39,40,37);(1,5,6,4)(2,7,8,3)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,38)(35,36)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=ghi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8831945);