#9224213

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	#39848	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4313 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 9131409);