#9210380

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=i,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9117576);