#8943074

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

4439 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,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, 8850270);