#8273410

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

4559 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)(11,16,13,17)(12,21,22,14)(19,20,23,24)(25,26)(27,35)(28,38)(29,40)(30,34)(31,37)(32,39)(33,36);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,17)(15,18)(19,24)(20,23)(21,22)(25,27)(26,31)(28,36)(29,39)(30,37)(32,33)(34,35)(38,40);(2,5)(6,8)(9,11,15,13)(10,16,18,17)(12,19,22,23)(14,24,21,20)(25,28,30,39)(26,32,34,38)(27,36,37,29)(31,33,35,40);(3,7)(6,8)(9,12)(10,14)(11,19)(13,23)(15,22)(16,24)(17,20)(18,21)(25,29)(26,33)(27,28)(30,36)(31,38)(32,35)(34,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,13,15,11)(10,17,18,16)(12,19,22,23)(14,24,21,20)(25,27)(26,31)(28,36)(29,39)(30,37)(32,33)(34,35)(38,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,12)(11,20)(13,24)(15,21)(16,23)(17,19)(18,22)(25,26)(27,31)(28,38)(29,33)(30,34)(32,39)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=gh,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, 8180606);