#8806886

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	#39443	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4336 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)(4,5)(6,8)(9,10,14,15)(11,16,19,13)(12,17,22,23)(18,24,20,21)(26,30)(27,35)(28,38)(34,40);(2,5)(6,7)(10,15)(13,16)(17,23)(21,24)(25,26,31,30)(27,36,35,32)(28,33,38,39)(29,40,37,34);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,16,15,13)(12,20,22,18)(17,21,23,24)(25,27,31,35)(26,32,30,36)(28,37,38,29)(33,40,39,34);(1,3,4,8)(2,6,5,7)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,22)(23,24)(25,28)(26,33)(27,37)(29,35)(30,39)(31,38)(32,40)(34,36);(9,12,14,22)(10,17,15,23)(11,18,19,20)(13,21,16,24)(25,29,31,37)(26,34,30,40)(27,28,35,38)(32,33,36,39);(9,13)(10,11)(12,21)(14,16)(15,19)(17,18)(20,23)(22,24)(25,30,31,26)(27,36,35,32)(28,39,38,33)(29,40,37,34)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714082);