#8467107

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

4598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,d^2=h,e^2=ghi,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, 8374303);