#8934664

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

4286 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)(11,16)(12,15)(13,22)(14,23)(17,20)(18,24)(19,21)(25,26)(27,28)(29,33)(30,38)(31,32)(34,39)(35,40)(36,37);(3,8)(5,6)(9,11,15,19)(10,16,12,21)(13,14,20,24)(17,18,22,23)(26,32)(28,33)(34,38)(36,40);(1,3)(2,5)(4,8)(6,7)(10,12)(11,19)(14,24)(17,22)(25,27,31,29)(26,33,32,28)(30,37,39,35)(34,36,38,40);(1,3,4,8)(2,6,7,5)(9,12,15,10)(11,16,19,21)(13,17,20,22)(14,23,24,18)(25,28,31,33)(26,27,32,29)(30,36,39,40)(34,35,38,37);(1,3)(2,6)(4,8)(5,7)(9,13,15,20)(10,17,12,22)(11,14,19,24)(16,18,21,23)(25,29,31,27)(26,28,32,33)(30,37,39,35)(34,36,38,40);(1,2)(3,6)(4,7)(5,8)(9,14)(10,18)(11,20)(12,23)(13,19)(15,24)(16,22)(17,21)(25,30,31,39)(26,34,32,38)(27,35,29,37)(28,36,33,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,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, 8841860);