#8751550

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

4285 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)(11,19)(12,18)(13,17)(14,20)(15,16)(21,24)(22,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,4,7)(2,5,6,8)(10,16)(12,22)(17,19)(20,21)(26,30)(31,33)(32,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,17,16,19)(12,20,22,21)(14,18,24,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3)(2,5)(4,7)(6,8)(9,12,15,22)(10,18,16,23)(11,21,13,20)(14,19,24,17)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,13,15,11)(10,19,16,17)(12,20,22,21)(14,18,24,23)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(9,14,15,24)(10,20,16,21)(11,18,13,23)(12,17,22,19)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=gi,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, 8658746);