#8383465

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

4448 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)(3,7)(9,10,15,12)(11,14,17,18)(13,16,23,20)(19,22,21,24)(26,28)(31,35)(32,36)(38,40);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,18)(13,21)(15,17)(16,24)(19,23)(20,22)(26,28)(27,33)(32,36)(34,39);(1,2)(3,6)(4,5)(7,8)(9,10,15,12)(11,18,17,14)(13,16,23,20)(19,24,21,22)(25,26,30,28)(27,31,33,35)(29,36,37,32)(34,40,39,38);(1,2)(3,6)(4,5)(7,8)(9,12,15,10)(11,18,17,14)(13,16,23,20)(19,22,21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,28,30,26)(27,31,33,35)(29,36,37,32)(34,38,39,40);(9,14,15,18)(10,17,12,11)(13,22,23,24)(16,21,20,19)(25,29,30,37)(26,32,28,36)(27,34,33,39)(31,38,35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8290661);