#10142188

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

4648 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=ghi,e^2=gh,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, 10049384);