#8433733

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

4708 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)(10,13)(11,12)(18,21)(19,20)(25,26)(27,37)(28,34)(29,31)(30,39)(32,36)(33,40)(35,38);(3,7)(6,8)(9,10,15,13)(11,17,12,16)(14,18,22,21)(19,24,20,23)(25,27)(26,33)(28,30)(29,38)(31,39)(32,40)(34,35)(36,37);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,28,32,29)(26,34,36,31)(27,30,40,38)(33,35,37,39);(1,3)(2,6)(4,7)(5,8)(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,29,32,28)(26,34,36,31)(27,38,40,30)(33,35,37,39);(1,2)(3,6)(4,5)(7,8)(9,13,15,10)(11,17,12,16)(14,21,22,18)(19,24,20,23)(25,30,32,38)(26,35,36,39)(27,28,40,29)(31,33,34,37);(9,14,15,22)(10,18,13,21)(11,19,12,20)(16,23,17,24)(25,31,32,34)(26,29,36,28)(27,39,40,35)(30,33,38,37)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8340929);