#8446093

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

4341 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)(3,4)(9,10)(11,20)(12,17)(13,23)(14,24)(15,19)(16,22)(18,21)(25,26,30,31)(27,33,36,32)(28,37,38,29)(34,35,40,39);(1,2,6,5)(3,8,4,7)(9,11)(10,16)(12,14)(13,21)(15,22)(17,18)(19,20)(23,24)(26,31)(28,38)(32,33)(34,40);(1,3,6,4)(2,7,5,8)(9,12,15,13)(10,17,19,23)(11,14,22,21)(16,18,20,24)(25,27,30,36)(26,32,31,33)(28,34,38,40)(29,35,37,39);(1,4,6,3)(2,8,5,7)(9,13,15,12)(10,17,19,23)(11,21,22,14)(16,18,20,24)(25,28,30,38)(26,29,31,37)(27,34,36,40)(32,35,33,39);(1,5,6,2)(3,8,4,7)(9,14,15,21)(10,18,19,24)(11,12,22,13)(16,17,20,23);(9,10)(11,16)(12,17)(13,23)(14,18)(15,19)(20,22)(21,24)(25,29)(26,28)(27,35)(30,37)(31,38)(32,34)(33,40)(36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=g,e^2=ghi,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, 8353289);