#8934668

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

4434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841864);