#8600856

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=gh,e^2=g,f^2=gh,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, 8508052);