#9712314

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

4433 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=ghi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,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, 9619510);