#8402683

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4209 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=ghi,e^2=h,f^2=gi,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, 8309879);