#8932736

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

4527 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839932);