#8393631

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	#35692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4684 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,30)(28,39)(29,40)(31,33)(32,38)(34,37)(35,36);(2,5)(6,7)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,27,31,38)(26,32,33,30)(28,36,37,29)(34,35,39,40);(1,2)(3,6)(4,5)(7,8)(12,20)(15,22)(18,24)(21,23)(26,33)(30,32)(34,39)(35,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28)(26,34)(27,36)(29,38)(30,40)(31,37)(32,35)(33,39);(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,29,31,36)(26,35,33,40)(27,37,38,28)(30,39,32,34);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,30)(26,27)(28,40)(29,39)(31,32)(33,38)(34,36)(35,37)

Polycyclic presentation:

2^9|b^2=gi,d^2=ghi,e^2=h,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, 8300827);