#9210372

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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)(9,10,14,17)(11,18,15,12)(13,16,23,24)(19,22,21,20)(25,26,31,29)(27,32,37,38)(28,34,30,33)(35,40,36,39);(1,2,6,8)(3,5,7,4)(9,11)(10,12)(13,21)(14,15)(16,22)(17,18)(19,23)(20,24)(25,27)(26,32)(28,35)(29,38)(30,36)(31,37)(33,39)(34,40);(1,3,6,7)(2,4,8,5)(9,12,14,18)(10,15,17,11)(13,20,23,22)(16,21,24,19)(25,28,31,30)(26,33,29,34)(27,35,37,36)(32,39,38,40);(1,4,6,5)(2,3,8,7)(9,12)(10,11)(13,22)(14,18)(15,17)(16,21)(19,24)(20,23);(1,2)(3,4)(5,7)(6,8)(9,13)(10,16)(11,19)(12,20)(14,23)(15,21)(17,24)(18,22)(25,29,31,26)(27,32,37,38)(28,34,30,33)(35,39,36,40);(1,5,6,4)(2,7,8,3)(9,11)(10,12)(13,21)(14,15)(16,22)(17,18)(19,23)(20,24)(25,30,31,28)(26,34,29,33)(27,36,37,35)(32,40,38,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=i,e^2=gh,f^2=ghi,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, 9117568);