#8902522

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

4647 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=i,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, 8809718);