#10040728

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

4153 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:

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

Polycyclic presentation:

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

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, 9947924);