#8639851

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

4431 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)(3,5,7,8)(9,10)(11,18)(12,13)(14,16)(15,17)(19,23)(20,21)(22,24)(25,26,32,30)(27,31,35,33)(28,34,29,39)(36,38,37,40);(3,7)(5,8)(11,12)(13,18)(19,20)(21,23)(25,27,32,35)(26,33,30,31)(28,36,29,37)(34,40,39,38);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,13,17,18)(14,19,22,20)(16,21,24,23)(25,28,32,29)(26,34,30,39)(27,36,35,37)(31,38,33,40);(1,3,4,7)(2,5,6,8)(9,12,15,11)(10,13,17,18)(14,19,22,20)(16,23,24,21)(25,29,32,28)(26,34,30,39)(27,37,35,36)(31,38,33,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,17)(14,21)(15,18)(16,19)(20,24)(22,23)(25,30,32,26)(27,33,35,31)(28,34,29,39)(36,38,37,40);(9,14,15,22)(10,16,17,24)(11,19,12,20)(13,21,18,23)(25,31,32,33)(26,35,30,27)(28,38,29,40)(34,37,39,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8547047);