#8533626

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

4734 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)(3,7)(9,10,16,20)(11,21,24,17)(12,18,13,14)(15,19,23,22)(25,26,29,30)(27,33,35,31)(28,32,37,36)(34,39,40,38);(1,2,4,5)(3,6,7,8)(9,11)(10,17)(12,19)(13,22)(14,23)(15,18)(16,24)(20,21)(26,30)(31,33)(32,36)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,20,14)(11,19,24,22)(15,21,23,17)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,2)(3,6)(4,5)(7,8)(9,13,16,12)(10,18,20,14)(11,22,24,19)(15,21,23,17)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,14,16,18)(10,12,20,13)(11,23,24,15)(17,19,21,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(9,15,16,23)(10,19,20,22)(11,18,24,14)(12,21,13,17)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)

Polycyclic presentation:

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

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, 8440822);