#8436551

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

4406 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=gh,e^2=g,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, 8343747);