#8339884

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

4415 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8247080);