#8517331

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

4500 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=g,e^2=g,f^2=g,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, 8424527);