#8687580

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8594776);