#8463223

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

4345 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8370419);