#8302014

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

4263 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)(11,20)(12,14)(13,23)(15,19)(16,22)(17,21)(18,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,19,20)(12,13,21,24)(14,18,17,23)(26,27)(31,33)(32,34)(38,39);(2,5)(6,8)(9,12,15,21)(10,17,19,14)(11,13,22,24)(16,23,20,18)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(3,7)(6,8)(9,11)(10,16)(12,13)(14,18)(15,22)(17,23)(19,20)(21,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,21)(12,22)(14,20)(15,24)(16,17)(19,23);(1,2)(3,6)(4,5)(7,8)(9,14,15,17)(10,12,19,21)(11,18,22,23)(13,20,24,16)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8209210);