#8599223

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

4564 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)(11,17)(12,14)(13,16)(15,19)(18,24)(20,22)(21,23)(25,26)(27,32)(28,29)(30,35)(31,33)(34,40)(36,37)(38,39);(2,5)(6,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,24)(18,22)(20,23)(25,27)(26,32)(28,38)(29,39)(30,33)(31,35)(34,36)(37,40);(1,2,4,5)(3,6,7,8)(13,22)(16,20)(18,21)(23,24)(26,33)(27,35)(29,40)(36,38);(3,7)(6,8)(9,12,14,10)(11,17,19,15)(13,20,22,16)(18,24,21,23)(25,28,31,34)(26,29,33,40)(27,36,35,38)(30,39,32,37);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)(25,29)(26,34)(27,37)(28,33)(30,36)(31,40)(32,38)(35,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,15)(13,18)(14,19)(16,24)(20,23)(21,22)(25,30)(26,35)(27,33)(28,37)(29,36)(31,32)(34,39)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506419);