#8496872

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

4576 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8404068);