#8463564

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

4448 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,e^2=ghi,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, 8370760);