#9712085

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	#40743	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4221 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)(3,4)(5,8)(6,7)(9,10,15,13)(11,17,12,16)(14,18,22,21)(19,24,20,23)(26,28)(27,30)(33,36)(35,37);(1,3)(2,6)(4,8)(5,7)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(1,4)(2,7)(3,8)(5,6)(9,10,15,13)(11,16,12,17)(14,18,22,21)(19,23,20,24)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,2,5,8)(3,6,7,4)(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(1,4)(2,3)(5,6)(7,8)(9,13,15,10)(11,17,12,16)(14,18,22,21)(19,23,20,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,3)(2,6)(4,8)(5,7)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9619281);