#9724841

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

4614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=i,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, 9632037);