#8934800

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

4452 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)(3,5)(4,8)(6,7)(9,10,15,17)(11,16,14,24)(12,13,22,18)(19,23,21,20)(25,26,31,37)(27,35,29,32)(28,34,39,33)(30,38,40,36);(1,3)(2,5)(4,6)(7,8)(9,11,15,14)(10,16,17,24)(12,21,22,19)(13,20,18,23)(25,27,31,29)(26,32,37,35)(28,38,39,36)(30,33,40,34);(1,2)(3,8)(4,5)(6,7)(10,17)(11,14)(12,22)(20,23)(25,28,31,39)(26,33,37,34)(27,38,29,36)(30,32,40,35);(2,7)(3,4)(9,12,15,22)(10,18,17,13)(11,19,14,21)(16,20,24,23)(25,28)(26,34)(27,36)(29,38)(30,35)(31,39)(32,40)(33,37);(1,4,6,3)(2,5,7,8)(9,13,15,18)(10,12,17,22)(11,20,14,23)(16,21,24,19)(25,29,31,27)(26,35,37,32)(28,36,39,38)(30,34,40,33);(1,5)(2,3)(4,7)(6,8)(9,14,15,11)(10,16,17,24)(12,21,22,19)(13,23,18,20)(25,30,31,40)(26,36,37,38)(27,33,29,34)(28,35,39,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841996);