#8798486

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	#48696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gi,e^2=hi,f^2=g,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, 8705682);