#9726080

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4263 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)(11,17)(12,21)(13,15)(14,18)(16,19)(20,22)(23,24)(25,26,31,35)(27,36,38,32)(28,39,40,33)(29,30,34,37);(1,3,5,7)(2,4,8,6)(9,11,16,13)(10,17,19,15)(12,20,14,23)(18,24,21,22)(25,27)(26,32)(28,37)(29,33)(30,40)(31,38)(34,39)(35,36);(1,3)(2,6)(4,8)(5,7)(9,12,16,14)(10,18,19,21)(11,20,13,23)(15,22,17,24)(25,28,31,40)(26,33,35,39)(27,37,38,30)(29,36,34,32);(1,2,5,8)(3,4,7,6)(9,13,16,11)(10,15,19,17)(12,20,14,23)(18,24,21,22)(25,29,31,34)(26,30,35,37)(27,33,38,39)(28,32,40,36);(1,3)(2,4)(5,7)(6,8)(9,14,16,12)(10,18,19,21)(11,20,13,23)(15,24,17,22)(25,30)(26,34)(27,28)(29,35)(31,37)(32,33)(36,39)(38,40);(1,4)(2,7)(3,8)(5,6)(9,15)(10,11)(12,22)(13,19)(14,24)(16,17)(18,20)(21,23)(25,27)(26,32)(28,37)(29,33)(30,40)(31,38)(34,39)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=di,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, 9633276);