#8904455

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

4446 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)(11,16)(12,14)(13,20)(15,19)(17,21)(18,22)(23,24)(25,26,30,35)(27,31,37,32)(28,33,38,39)(29,36,40,34);(1,2)(3,6)(4,7)(5,8)(9,11,14,19)(10,15,12,16)(13,18,21,24)(17,22,20,23)(25,27)(26,31)(28,34)(29,33)(30,37)(32,35)(36,38)(39,40);(1,3)(2,5)(4,8)(6,7)(10,12)(13,21)(15,16)(18,24)(25,27,30,37)(26,32,35,31)(28,36,38,34)(29,39,40,33);(3,8)(5,6)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,27)(26,31)(28,34)(29,33)(30,37)(32,35)(36,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28)(26,33)(27,36)(29,32)(30,38)(31,40)(34,37)(35,39);(1,2)(3,6)(4,7)(5,8)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29,30,40)(26,34,35,36)(27,33,37,39)(28,32,38,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8811651);