#8750840

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

4524 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)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,26)(27,32)(28,38)(29,33)(30,34)(31,35)(36,40)(37,39);(3,7)(6,8)(10,14)(15,17)(16,20)(23,24)(25,27,31,28)(26,32,35,38)(29,36,34,39)(30,37,33,40);(1,2)(3,6)(4,5)(7,8)(10,14)(15,17)(16,20)(23,24)(27,28)(32,38)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,28,31,27)(26,32,35,38)(29,39,34,36)(30,37,33,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,29)(26,33)(27,36)(28,39)(30,35)(31,34)(32,40)(37,38);(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,30,31,33)(26,34,35,29)(27,37,28,40)(32,39,38,36)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658036);