#9305975

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

4319 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=gh,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=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9213171);