#8726394

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	#39402	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4427 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633590);