#10407830

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

4551 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10315026);