#9726450

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

4559 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=gi,f^2=h,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, 9633646);