#8463301

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

4433 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)(9,10,14,18)(11,13,17,15)(12,16,21,24)(19,20,23,22)(26,30)(31,36)(32,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,17)(10,13,18,15)(12,19,21,23)(16,20,24,22)(27,33)(31,36)(34,39)(38,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2,4,6)(3,5,8,7)(9,11)(10,13)(12,19)(14,17)(15,18)(16,20)(21,23)(22,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,21)(10,16,18,24)(11,19,17,23)(13,20,15,22)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,18)(12,20)(14,15)(16,23)(19,24)(21,22)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370497);