#8936774

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

4396 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8843970);