#8934364

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

4568 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)(3,7)(4,6)(5,8)(9,10,14,17)(11,18,21,15)(12,13,16,23)(19,20,22,24)(25,26,29,33)(27,31,35,30)(28,36,38,32)(34,39,37,40);(1,2,4,6)(3,5,8,7)(9,11,14,21)(10,15,17,18)(12,22,16,19)(13,20,23,24)(25,27,29,35)(26,30,33,31)(28,34,38,37)(32,39,36,40);(1,2)(3,5)(4,6)(7,8)(12,16)(13,23)(19,22)(20,24)(25,27,29,35)(26,30,33,31)(28,37,38,34)(32,40,36,39);(1,3,4,8)(2,5,6,7)(9,12,14,16)(10,13,17,23)(11,19,21,22)(15,24,18,20)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(9,11)(10,15)(12,22)(13,20)(14,21)(16,19)(17,18)(23,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(9,13)(10,16)(11,20)(12,17)(14,23)(15,19)(18,22)(21,24)(25,28,29,38)(26,32,33,36)(27,34,35,37)(30,39,31,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=gh,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, 8841560);