#8592010

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

4587 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8499206);