#8688196

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

4331 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,21)(13,17)(14,15)(16,18)(20,22)(23,24)(25,26)(27,34)(28,32)(29,33)(30,35)(31,36)(37,40)(38,39);(1,2,4,6)(3,5,8,7)(9,11,16,15)(10,14,18,19)(12,20,13,23)(17,22,21,24)(25,27)(26,30)(28,37)(29,38)(31,35)(32,33)(34,36)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12,16,13)(10,17,18,21)(11,20,15,23)(14,22,19,24)(25,28,31,38)(26,32,36,39)(27,37,35,29)(30,33,34,40);(1,3,4,8)(2,5,6,7)(9,13,16,12)(10,17,18,21)(11,20,15,23)(14,24,19,22)(25,29,31,37)(26,33,36,40)(27,38,35,28)(30,32,34,39);(9,14,16,19)(10,11,18,15)(12,22,13,24)(17,20,21,23)(25,27)(26,34)(28,37)(29,38)(30,36)(31,35)(32,40)(33,39);(9,15,16,11)(10,14,18,19)(12,23,13,20)(17,22,21,24)(25,30,31,34)(26,35,36,27)(28,33,38,40)(29,39,37,32)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=g,e^2=h,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, 8595392);