#8942943

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

4276 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:

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

Polycyclic presentation:

2^9|a^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, 8850139);