#8943033

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	#50642	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8850229);