#9930794

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	#51746	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4177 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)(5,6)(7,8)(9,10)(11,18)(12,17)(13,22)(14,16)(15,20)(19,24)(21,23)(26,30)(31,33)(32,38)(39,40);(2,6)(3,7)(9,11)(10,15)(12,19)(13,17)(14,20)(16,18)(21,22)(23,24)(25,26,29,30)(27,33,35,31)(28,32,37,38)(34,40,36,39);(1,3,5,7)(2,4,6,8)(10,16)(12,21)(15,18)(19,22);(1,4)(2,7)(3,6)(5,8)(9,12,14,21)(10,17,16,23)(11,19,20,22)(13,18,24,15);(9,11)(10,18)(12,19)(13,23)(14,20)(15,16)(17,24)(21,22)(25,27,29,35)(26,31,30,33)(28,36,37,34)(32,40,38,39);(9,13,14,24)(10,19,16,22)(11,17,20,23)(12,18,21,15)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9837990);