#8688342

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

4357 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=hi,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, 8595538);