#9712235

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

4376 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)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26,31,35)(27,32,30,34)(28,29,33,39)(36,40,38,37);(1,2,5,4)(3,6,8,7)(11,17)(14,21)(18,23)(22,24)(25,27,31,30)(26,32,35,34)(28,38,33,36)(29,37,39,40);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,28,31,33)(26,29,35,39)(27,36,30,38)(32,40,34,37);(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24);(1,3)(2,7)(4,6)(5,8)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,29)(26,33)(27,37)(28,35)(30,40)(31,39)(32,36)(34,38);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,37,39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=g,e^2=gh,f^2=gi,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, 9619431);