#8549628

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

4597 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8456824);