#8754850

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

4406 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=i,e^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, 8662046);