#8893456

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

4666 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,e^2=g,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, 8800652);