#8273411

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

4302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=gh,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, 8180607);