#8862322

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

4573 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=i,f^2=g,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, 8769518);