#9224212

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	#37810	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 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)(11,16)(12,14)(13,17)(15,19)(18,24)(20,22)(21,23)(27,35)(30,40)(32,36)(34,37);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,23)(20,24)(21,22)(25,26,31,28)(27,36,35,32)(29,38,39,33)(30,34,40,37);(1,3)(2,6)(4,7)(5,8)(25,27)(26,32)(28,36)(29,37)(30,33)(31,35)(34,39)(38,40);(1,4,5,6)(2,3,7,8)(9,12,14,10)(11,16,19,15)(13,17,22,20)(18,23,21,24)(25,28,31,26)(27,36,35,32)(29,33,39,38)(30,34,40,37);(1,3)(2,4)(5,8)(6,7)(9,11)(10,16)(12,15)(13,21)(14,19)(17,23)(18,22)(20,24)(25,29,31,39)(26,33,28,38)(27,37,35,34)(30,36,40,32);(1,2)(3,6)(4,8)(5,7)(9,13,14,22)(10,17,12,20)(11,18,19,21)(15,23,16,24)(25,30,31,40)(26,34,28,37)(27,33,35,38)(29,32,39,36)

Polycyclic presentation:

2^9|b^2=h,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, 9131408);