#9724833

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

4234 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=gh,f^2=g,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, 9632029);