#8159334

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

4730 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)(9,10,14,15)(11,12,18,16)(13,21,22,17)(19,23,24,20)(25,26)(27,32)(28,34)(29,38)(30,37)(31,33)(35,39)(36,40);(3,7)(6,8)(10,15)(11,18)(17,21)(19,24)(25,27)(26,32)(28,31)(29,37)(30,38)(33,34)(35,40)(36,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(17,21)(19,24)(26,33)(27,28)(35,38)(36,37);(1,3)(2,6)(4,7)(5,8)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,28,31,27)(26,34,33,32)(29,37,39,36)(30,35,40,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29)(26,35)(27,37)(28,36)(30,34)(31,39)(32,40)(33,38);(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,30)(26,36)(27,38)(28,35)(29,34)(31,40)(32,39)(33,37)

Polycyclic presentation:

2^9|a^2=g,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066530);