#8821734

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

4358 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)(3,4)(5,6)(7,8)(25,26,30,33)(27,29,36,39)(28,32,38,31)(34,40,35,37);(2,6)(4,7)(9,10,14,12)(11,15,17,18)(13,20,21,16)(19,24,23,22)(25,27,30,36)(26,29,33,39)(28,35,38,34)(31,40,32,37);(2,6)(3,8)(10,12)(11,17)(16,20)(19,23)(25,28,30,38)(26,31,33,32)(27,34,36,35)(29,37,39,40);(1,3,5,8)(2,7,6,4)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,33)(34,36)(39,40);(1,2)(3,7)(4,8)(5,6)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,27)(26,29)(28,35)(30,36)(31,40)(32,37)(33,39)(34,38);(1,4,5,7)(2,8,6,3)(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,29)(26,27)(28,37)(30,39)(31,34)(32,35)(33,36)(38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8728930);