#8943327

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	C₂×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=gh,e^2=h,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, 8850523);