#8458689

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

4594 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,14)(13,23)(15,19)(16,24)(17,22)(18,21)(25,26)(27,31)(28,38)(29,33)(30,34)(32,35)(36,40)(37,39);(2,5)(6,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,24)(17,18)(19,20)(25,27,32,30)(26,31,35,34)(28,37,29,36)(33,40,38,39);(1,2)(3,6)(4,5)(7,8)(25,28,32,29)(26,33,35,38)(27,36,30,37)(31,39,34,40);(1,3,4,8)(2,6,5,7)(9,12,15,22)(10,17,19,14)(11,21,24,13)(16,18,20,23)(25,29,32,28)(26,33,35,38)(27,37,30,36)(31,39,34,40);(9,13,15,21)(10,18,19,23)(11,22,24,12)(14,16,17,20)(25,30,32,27)(26,34,35,31)(28,37,29,36)(33,40,38,39);(9,14,15,17)(10,12,19,22)(11,23,24,18)(13,16,21,20)(25,31,32,34)(26,27,35,30)(28,39,29,40)(33,36,38,37)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=gh,e^2=gi,f^2=gi,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, 8365885);