#8934463

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=g,e^2=gh,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, 8841659);