#8393429

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	#43011	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4456 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,22)(14,20)(15,17)(19,23)(21,24)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(2,5)(6,8)(9,11,15,12)(10,16,17,18)(13,21,23,20)(14,19,24,22)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(3,7)(6,8)(10,17)(14,24)(16,18)(19,22)(26,30)(31,38)(32,35)(39,40);(1,2,4,5)(3,6,7,8)(9,12,15,11)(10,18,17,16)(13,20,23,21)(14,22,24,19)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,2)(3,6)(4,5)(7,8)(9,13,15,23)(10,19,17,22)(11,20,12,21)(14,18,24,16)(25,28,29,37)(26,32,30,35)(27,33,34,36)(31,39,38,40);(1,3,4,7)(2,6,5,8)(9,14,15,24)(10,20,17,21)(11,19,12,22)(13,18,23,16)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300625);