#10181036

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

4255 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10088232);