#8873696

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

4496 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,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, 8780892);