#8750322

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

4461 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)(9,10,14,15)(11,16,19,22)(12,21,23,17)(13,18,24,20)(25,26,31,32)(27,29,28,35)(30,33,39,40)(34,37,36,38);(3,7)(5,8)(10,15)(11,19)(17,21)(18,20)(25,27)(26,29)(28,31)(30,34)(32,35)(33,37)(36,39)(38,40);(1,2)(3,5)(4,6)(7,8)(10,15)(11,19)(17,21)(18,20)(26,32)(27,28)(33,40)(34,36);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,24)(18,23)(25,28,31,27)(26,29,32,35)(30,34,39,36)(33,38,40,37);(1,2)(3,5)(4,6)(7,8)(9,12,14,23)(10,17,15,21)(11,20,19,18)(13,16,24,22)(25,29)(26,27)(28,32)(30,38)(31,35)(33,36)(34,40)(37,39);(9,13,14,24)(10,18,15,20)(11,17,19,21)(12,22,23,16)(25,30,31,39)(26,33,32,40)(27,34,28,36)(29,37,35,38)

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8657518);