#8692624

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

4649 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8599820);