#8436583

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

4446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=h,f^2=g,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, 8343779);