#8156940

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

4370 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=gh,e^2=g,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, 8064136);