#8284197

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

4630 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,f^2=g,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, 8191393);