#8107267

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

4558 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014463);