#8097276

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

4591 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004472);