#9712217

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

4276 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=gh,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, 9619413);