#8943986

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

4386 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8851182);