#9713478

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

4491 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)(3,4)(5,8)(6,7)(9,10,15,17)(11,14,12,16)(13,18,22,24)(19,21,20,23)(25,26,32,34)(27,35,29,33)(28,38,30,31)(36,37,39,40);(1,3)(2,4)(5,6)(7,8)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,21,24,23)(25,27,32,29)(26,33,34,35)(28,36,30,39)(31,37,38,40);(1,2,7,6)(3,5,8,4)(10,17)(11,12)(18,24)(19,20)(25,28,32,30)(26,31,34,38)(27,36,29,39)(33,37,35,40);(1,4,7,5)(2,8,6,3)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,21,24,23)(25,29,32,27)(26,33,34,35)(28,36,30,39)(31,40,38,37);(1,5,7,4)(2,8,6,3)(9,13,15,22)(10,18,17,24)(11,19,12,20)(14,21,16,23)(25,30,32,28)(26,31,34,38)(27,36,29,39)(33,40,35,37);(1,6,7,2)(3,5,8,4)(9,14)(10,11)(12,17)(13,21)(15,16)(18,19)(20,24)(22,23)(25,31)(26,30)(27,37)(28,34)(29,40)(32,38)(33,39)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=h,e^2=h,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, 9620674);