#8754890

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

4187 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662086);