#8228681

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

4627 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:

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

Polycyclic presentation:

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

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, 8135877);