#8496588

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

4513 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8403784);