#8108774

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

4406 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=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, 8015970);