#8900155

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

4507 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)(3,5,7,8)(9,10,14,16)(11,15,20,19)(12,17,23,22)(13,18,24,21)(25,26,31,32)(27,34,28,33)(29,35,39,37)(30,38,40,36);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,21)(13,17)(14,20)(16,19)(18,23)(22,24)(26,32)(29,39)(33,34)(36,38);(1,2)(3,5)(4,6)(7,8)(10,16)(11,20)(17,22)(18,21)(25,27,31,28)(26,33,32,34)(29,36,39,38)(30,35,40,37);(9,12)(10,17)(11,21)(13,15)(14,23)(16,22)(18,20)(19,24)(25,28,31,27)(26,34,32,33)(29,36,39,38)(30,35,40,37);(1,3)(2,5)(4,7)(6,8)(9,13,14,24)(10,18,16,21)(11,22,20,17)(12,19,23,15)(25,29,31,39)(26,35,32,37)(27,36,28,38)(30,33,40,34);(1,2)(3,5)(4,6)(7,8)(9,11)(10,19)(12,21)(13,22)(14,20)(15,16)(17,24)(18,23)(25,30)(26,36)(27,37)(28,35)(29,33)(31,40)(32,38)(34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=g,e^2=gi,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, 8807351);