#9713768

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

4365 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,16,19,12)(13,21,23,17)(18,20,22,24)(28,36)(32,39)(35,37)(38,40);(3,8)(6,7)(9,11,14,19)(10,12,15,16)(13,18,23,22)(17,20,21,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(1,3)(2,6)(4,7)(5,8)(10,15)(12,16)(17,21)(20,24)(25,27,30,29)(26,31,33,34)(28,35,36,37)(32,38,39,40);(1,2,5,4)(3,7,8,6)(9,12,14,16)(10,11,15,19)(13,20,23,24)(17,18,21,22)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,16)(12,15)(13,22)(14,19)(17,20)(18,23)(21,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,18)(12,20)(14,23)(15,21)(16,24)(19,22)(25,29,30,27)(26,31,33,34)(28,37,36,35)(32,38,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,f^2=i,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, 9620964);