#8750276

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

4325 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657472);