#8607460

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

4404 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8514656);