#8526805

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

4346 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8434001);