#9034331

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

4270 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)(6,8)(9,10,15,13)(11,16,14,18)(12,17,21,22)(19,23,20,24)(25,26)(27,36)(28,29)(30,38)(31,32)(33,37)(34,40)(35,39);(3,7)(6,8)(9,11,15,14)(10,16,13,18)(12,20,21,19)(17,24,22,23)(26,32)(27,37)(29,34)(35,38);(1,2)(3,6)(4,5)(7,8)(9,12,15,21)(10,17,13,22)(11,19,14,20)(16,23,18,24)(25,27)(26,33)(28,38)(29,39)(30,34)(31,37)(32,36)(35,40);(9,13,15,10)(11,18,14,16)(12,17,21,22)(19,23,20,24)(25,28)(26,34)(27,38)(29,32)(30,33)(31,40)(35,37)(36,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,22)(14,19)(15,21)(16,24)(18,23)(25,29,31,34)(26,28,32,40)(27,39,37,30)(33,38,36,35);(1,2)(3,6)(4,5)(7,8)(9,14,15,11)(10,18,13,16)(12,19,21,20)(17,23,22,24)(25,30,31,39)(26,35,32,38)(27,34,37,29)(28,33,40,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=gi,e^2=h,f^2=i,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, 8941527);