#8942944

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

4294 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)(11,17)(13,21)(20,23)(26,30)(31,36)(32,37)(38,40);(1,2)(3,5)(4,6)(7,8)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20)(28,35)(32,37)(33,39)(38,40);(1,2,4,6)(3,7,8,5)(9,10,14,15)(11,16,17,12)(13,22,21,18)(19,20,24,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(3,8)(5,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,28)(26,32)(27,33)(29,35)(30,37)(31,38)(34,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,21)(10,18,15,22)(11,19,17,24)(12,20,16,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=gh,e^2=g,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, 8850140);