#9232814

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

4433 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)(10,11)(15,17)(16,18)(23,24)(25,26)(27,28)(29,37)(30,33)(31,35)(32,40)(34,38)(36,39);(1,3,4,7)(2,5,6,8)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(25,27,30,35)(26,28,33,31)(29,34,40,39)(32,36,37,38);(1,2)(3,5)(4,6)(7,8)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,37)(39,40);(9,11,14,10)(12,15,20,17)(13,16,22,18)(19,24,21,23)(25,28,30,31)(26,27,33,35)(29,38,40,36)(32,39,37,34);(1,3)(2,5)(4,7)(6,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,27)(26,28)(29,39)(30,35)(31,33)(32,38)(34,40)(36,37);(1,2)(3,5)(4,6)(7,8)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29,30,40)(26,32,33,37)(27,34,35,39)(28,36,31,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gi,e^2=g,f^2=gi,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, 9140010);