#8899309

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	#35520	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,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^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, 8806505);