#8934316

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	#35567	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4398 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=gh,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, 8841512);