#8889223

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

4564 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)(9,10)(11,12)(13,21)(14,18)(15,16)(17,22)(19,23)(20,24)(25,26,31,36)(27,34,29,32)(28,33,39,37)(30,38,40,35);(3,7)(6,8)(9,10,14,18)(11,15,16,12)(13,21,22,17)(19,20,24,23)(27,29)(30,40)(32,34)(35,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,27,31,29)(26,32,36,34)(28,38,39,35)(30,37,40,33);(9,11)(10,15)(12,18)(13,19)(14,16)(17,23)(20,21)(22,24)(25,28)(26,33)(27,35)(29,38)(30,34)(31,39)(32,40)(36,37);(1,3)(2,6)(4,7)(5,8)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,29,31,27)(26,34,36,32)(28,35,39,38)(30,33,40,37);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,30,31,40)(26,35,36,38)(27,37,29,33)(28,34,39,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796419);