#8899337

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

4457 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806533);