#8436842

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

4393 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=g,e^2=gh,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, 8344038);