#8799468

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

4410 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8706664);