#8722234

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	#39953	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4482 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8629430);