#8174393

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

4339 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)(9,10,15,19)(11,16,21,24)(12,13,17,22)(14,23,20,18)(25,26)(27,37)(28,34)(29,30)(31,39)(32,36)(33,40)(35,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,21)(10,16,19,24)(12,20,17,14)(13,18,22,23)(25,27)(26,33)(28,35)(29,38)(30,39)(31,34)(32,40)(36,37);(1,2)(3,6)(4,5)(7,8)(9,12,15,17)(10,13,19,22)(11,14,21,20)(16,23,24,18)(25,28,32,30)(26,34,36,29)(27,35,40,39)(31,37,38,33);(2,5)(6,7)(9,12)(10,13)(11,20)(14,21)(15,17)(16,18)(19,22)(23,24)(25,29,32,34)(26,28,36,30)(27,38,40,31)(33,35,37,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,18)(12,19)(14,16)(15,22)(20,24)(21,23)(25,30,32,28)(26,29,36,34)(27,39,40,35)(31,33,38,37);(1,3,4,8)(2,6,5,7)(9,14,15,20)(10,18,19,23)(11,12,21,17)(13,16,22,24)(25,31,32,38)(26,35,36,39)(27,34,40,29)(28,37,30,33)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,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, 8081589);