#10407755

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

4522 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)(3,8)(4,7)(5,6)(9,10,15,12)(11,16,18,13)(14,17,23,20)(19,24,22,21)(25,26,31,33)(27,32,36,40)(28,29,34,38)(30,37,39,35);(1,3,6,4)(2,7,5,8)(9,11)(10,16)(12,13)(14,22)(15,18)(17,21)(19,23)(20,24)(25,27,31,36)(26,32,33,40)(28,35,34,37)(29,30,38,39);(1,2,6,5)(3,7,4,8)(11,18)(13,16)(19,22)(21,24)(26,33)(27,36)(29,38)(35,37);(1,2,6,5)(3,8,4,7)(9,12,15,10)(11,16,18,13)(14,20,23,17)(19,24,22,21)(25,28,31,34)(26,29,33,38)(27,37,36,35)(30,32,39,40);(1,4,6,3)(2,7,5,8)(9,13,15,16)(10,11,12,18)(14,21,23,24)(17,19,20,22)(25,29)(26,34)(27,30)(28,33)(31,38)(32,37)(35,40)(36,39);(1,5,6,2)(3,7,4,8)(9,14,15,23)(10,17,12,20)(11,19,18,22)(13,21,16,24)(25,30)(26,35)(27,29)(28,32)(31,39)(33,37)(34,40)(36,38)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=gi,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, 10314951);