#8754942

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) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=gi,e^2=gh,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, 8662138);