#9015679

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

4414 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)(9,10,14,18)(11,13,17,15)(12,16,22,20)(19,21,24,23)(25,26)(27,36)(28,30)(29,33)(31,35)(32,37)(34,40)(38,39);(1,2,4,6)(3,5,7,8)(9,11,14,17)(10,15,18,13)(12,19,22,24)(16,23,20,21)(27,37)(29,38)(32,36)(33,39);(1,3)(2,5)(4,7)(6,8)(9,10,14,18)(11,15,17,13)(12,20,22,16)(19,21,24,23)(25,27,31,37)(26,32,35,36)(28,29,34,38)(30,39,40,33);(9,10)(11,15)(12,20)(13,17)(14,18)(16,22)(19,21)(23,24)(25,28,31,34)(26,30,35,40)(27,38,37,29)(32,33,36,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,29,31,38)(26,33,35,39)(27,28,37,34)(30,32,40,36);(1,3)(2,5)(4,7)(6,8)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,24,20,19)(25,30)(26,34)(27,33)(28,35)(29,32)(31,40)(36,38)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8922875);