#8156929

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

4454 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)(11,17)(12,15)(13,16)(14,20)(18,22)(19,23)(21,24)(26,30)(27,33)(32,38)(34,35);(1,2,6,5)(3,8,4,7)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(1,3,6,4)(2,7,5,8)(10,12)(16,17)(18,20)(23,24)(27,33)(31,37)(34,35)(39,40);(1,4,6,3)(2,8,5,7)(9,12,15,10)(11,16,13,17)(14,20,22,18)(19,23,21,24)(25,27)(26,31)(28,35)(29,33)(30,37)(32,40)(34,36)(38,39);(1,5,6,2)(3,8,4,7)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40);(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8064125);