#9717749

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

4326 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=g,f^2=gh,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, 9624945);