#8651872

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

4377 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,e^2=i,f^2=gi,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, 8559068);