#8901578

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

4437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=ghi,e^2=gi,f^2=h,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, 8808774);