#8720356

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

4479 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627552);