#9232804

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

4541 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)(3,5)(4,8)(6,7)(9,10,15,19)(11,17,12,16)(13,18,24,14)(20,21,22,23);(2,7)(3,8)(9,11,15,12)(10,16,19,17)(13,20,24,22)(14,21,18,23)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(1,3,6,8)(2,4,7,5)(25,27)(26,31)(28,33)(29,34)(30,35)(32,37)(36,40)(38,39);(1,4,6,5)(2,3,7,8)(9,12,15,11)(10,17,19,16)(13,20,24,22)(14,21,18,23)(25,28,30,31)(26,27,33,35)(29,37,39,36)(32,38,40,34);(1,2)(3,4)(5,8)(6,7)(9,13,15,24)(10,18,19,14)(11,20,12,22)(16,23,17,21)(25,27)(26,28)(29,38)(30,35)(31,33)(32,37)(34,39)(36,40);(1,5,6,4)(2,8,7,3)(9,14,15,18)(10,13,19,24)(11,21,12,23)(16,20,17,22)(25,29,30,39)(26,32,33,40)(27,34,35,38)(28,36,31,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=ghi,e^2=gh,f^2=ghi,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, 9140000);