#8390894

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	#46099	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4450 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gi,e^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, 8298090);