#9713468

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

4742 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=h,e^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, 9620664);