#8215500

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	#38836	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4328 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)(11,16)(12,18)(13,20)(14,19)(15,17)(21,24)(22,23)(25,26,31,30)(27,34,32,28)(29,37,38,33)(35,36,39,40);(1,2)(3,7)(4,5)(6,8)(9,11)(10,16)(12,14)(13,17)(15,20)(18,19)(21,22)(23,24)(25,27,31,32)(26,28,30,34)(29,35,38,39)(33,36,37,40);(2,5)(6,7)(10,17)(11,20)(14,21)(18,23)(25,28,31,34)(26,32,30,27)(29,36,38,40)(33,39,37,35);(9,12)(10,18)(11,21)(13,19)(14,20)(15,22)(16,24)(17,23)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,3,4,8)(2,6,5,7)(9,13)(10,11)(12,19)(14,23)(15,16)(17,20)(18,21)(22,24)(25,29,31,38)(26,33,30,37)(27,35,32,39)(28,36,34,40);(1,2)(3,6)(4,5)(7,8)(9,14,15,21)(10,19,17,24)(11,12,20,22)(13,23,16,18)(25,30,31,26)(27,34,32,28)(29,37,38,33)(35,40,39,36)

Polycyclic presentation:

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

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, 8122696);