#8214384

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

4272 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)(11,21,23,17)(12,13,14,18)(15,19,22,24)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,2)(3,7)(4,5)(6,8)(9,11)(10,17)(12,19)(13,15)(14,24)(16,23)(18,22)(20,21)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(2,5)(6,7)(9,12,16,14)(10,18,20,13)(11,19,23,24)(15,17,22,21)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(9,13)(10,12)(11,22)(14,20)(15,23)(16,18)(17,24)(19,21)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,3,4,8)(2,6,5,7)(9,14,16,12)(10,13,20,18)(11,19,23,24)(15,17,22,21);(1,2)(3,6)(4,5)(7,8)(9,15,16,22)(10,19,20,24)(11,13,23,18)(12,21,14,17)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40)

Polycyclic presentation:

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

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, 8121580);