#9224208

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gi,e^2=h,f^2=gh,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, 9131404);