#9163976

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4273 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9071172);