#8934755

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	#47943	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,3)(2,5)(4,6)(7,8)(10,15)(11,18)(17,21)(19,24)(25,26,29,30)(27,31,35,33)(28,32,37,36)(34,38,40,39);(1,2)(3,8)(4,5)(6,7)(10,15)(11,18)(13,22)(20,23)(26,30)(28,37)(31,33)(34,40);(2,7)(3,4)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,4,6,3)(2,5,7,8)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,5)(2,3)(4,7)(6,8)(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^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, 8841951);