#8743451

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

4679 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gi,e^2=g,f^2=g,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, 8650647);