#8159242

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

4469 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)(11,13)(12,20)(14,15)(16,22)(17,18)(19,24)(21,23)(25,26,30,33)(27,34,31,28)(29,37,38,32)(35,36,39,40);(2,5)(6,7)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,27,30,31)(26,28,33,34)(29,35,38,39)(32,36,37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,20)(19,23)(25,27,30,31)(26,28,33,34)(29,35,38,39)(32,36,37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,20)(13,17)(14,15)(16,22)(19,21)(23,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,16,15,20)(11,19,17,23)(13,21,18,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,39)(35,37)(38,40);(9,13,14,18)(10,17,15,11)(12,21,22,24)(16,23,20,19)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8066438);