#9065738

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

4498 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8972934);