#8934783

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4273 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)(4,8)(6,7)(9,10)(11,16)(12,18)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26,31,35)(27,32,39,40)(28,30,37,33)(29,38,36,34);(1,3)(2,5)(4,6)(7,8)(10,15)(11,18)(17,21)(19,24)(25,27,31,39)(26,32,35,40)(28,29,37,36)(30,38,33,34);(1,2)(3,8)(4,5)(6,7)(10,15)(11,18)(13,22)(20,23)(25,28,31,37)(26,33,35,30)(27,36,39,29)(32,38,40,34);(2,7)(3,4)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,28)(26,33)(27,36)(29,39)(30,35)(31,37)(32,38)(34,40);(1,4,6,3)(2,5,7,8)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29,31,36)(26,34,35,38)(27,37,39,28)(30,40,33,32);(1,5)(2,3)(4,7)(6,8)(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,30,31,33)(26,28,35,37)(27,38,39,34)(29,40,36,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,e^2=gi,f^2=h,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, 8841979);