#9006981

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

4692 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8914177);