#9206758

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	#41057	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4292 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)(3,5)(4,7)(6,8)(9,10)(11,20)(12,18)(13,19)(14,16)(15,17)(21,22)(23,24)(26,30)(31,36)(32,37)(38,40);(1,2,6,8)(3,5,7,4)(9,11,15,14)(10,16,17,20)(12,19,22,23)(13,18,24,21)(28,35)(32,37)(33,39)(38,40);(1,3,6,7)(2,4,8,5)(10,17)(12,22)(16,20)(19,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(1,4,6,5)(2,3,8,7)(9,12,15,22)(10,18,17,21)(11,19,14,23)(13,16,24,20)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,24)(10,19,17,23)(11,21,14,18)(12,20,22,16)(25,28)(26,32)(27,33)(29,35)(30,37)(31,38)(34,39)(36,40);(1,5,6,4)(2,7,8,3)(9,14,15,11)(10,20,17,16)(12,23,22,19)(13,21,24,18)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113954);