#9718203

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	#2733	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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)(3,7)(4,6)(5,8)(10,15)(11,18)(17,23)(19,20)(26,30)(31,37)(32,38)(39,40);(1,2,4,6)(3,5,8,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26,29,30)(27,31,34,37)(28,32,36,38)(33,39,35,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,21,23,24);(1,3)(2,5)(4,8)(6,7)(9,12,14,22)(10,17,15,23)(11,19,18,20)(13,21,16,24);(9,13)(10,11)(12,21)(14,16)(15,18)(17,19)(20,23)(22,24)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=ghi,f^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, 9625399);