#8436531

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

4485 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,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, 8343727);