#8599172

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

4446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=h,f^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, 8506368);