#8873699

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) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8780895);