#8750831

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

4333 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)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(25,26)(27,35)(28,29)(30,32)(31,33)(34,39)(36,37)(38,40);(1,3,4,8)(2,5,6,7)(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,27,31,30)(26,32,33,35)(28,38,34,36)(29,37,39,40);(1,3)(2,5)(4,8)(6,7)(12,20)(17,22)(19,21)(23,24)(26,33)(28,34)(32,35)(36,38);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,28,31,34)(26,29,33,39)(27,36,30,38)(32,40,35,37);(1,2)(3,5)(4,6)(7,8)(9,13,15,11)(10,16,14,18)(12,21,20,19)(17,23,22,24)(25,29)(26,34)(27,37)(28,33)(30,40)(31,39)(32,36)(35,38);(9,14,15,10)(11,18,13,16)(12,17,20,22)(19,23,21,24)(25,30,31,27)(26,32,33,35)(28,36,34,38)(29,37,39,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=hi,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, 8658027);