#9210492

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
prime radical	#51410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9117688);