#8127454

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

4432 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034650);