#8421828

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

4621 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8329024);