#9434195

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	#40603	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=hi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9341391);