#9729223

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4364 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)(11,20)(12,17)(13,16)(14,18)(15,19)(21,22)(23,24)(25,26)(27,28)(29,33)(30,32)(31,35)(34,36)(37,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,19,20)(12,21,18,23)(14,22,17,24)(25,27)(26,31)(28,32)(29,37)(30,35)(33,36)(34,38)(39,40);(1,2,5,8)(3,4,7,6)(26,32)(28,31)(33,40)(36,39);(1,3)(2,6)(4,8)(5,7)(9,12,15,18)(10,17,19,14)(11,21,13,23)(16,24,20,22)(25,28,30,31)(26,27,32,35)(29,36,38,39)(33,34,40,37);(1,3)(2,4)(5,7)(6,8)(9,13,15,11)(10,16,19,20)(12,23,18,21)(14,22,17,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37);(1,4)(2,7)(3,8)(5,6)(9,14,15,17)(10,18,19,12)(11,22,13,24)(16,23,20,21)(25,27)(26,31)(28,32)(29,37)(30,35)(33,36)(34,38)(39,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,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, 9636419);