#8739451

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

4194 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gh,e^2=gh,f^2=gh,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, 8646647);