#8581530

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

4553 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488726);