#8934281

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

4500 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=h,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, 8841477);