#9030069

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
prime radical	#53431	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4471 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=h,e^2=h,f^2=gi,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, 8937265);