#8720359

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

4469 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)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,23,22,24)(25,26,31,36)(27,37,39,32)(28,30,35,33)(29,40,38,34);(1,2,6,5)(3,8,7,4)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,24)(21,22)(25,27)(26,32)(28,29)(30,34)(31,39)(33,40)(35,38)(36,37);(1,3)(2,4)(5,8)(6,7)(12,21)(16,20)(18,22)(23,24)(25,28,31,35)(26,33,36,30)(27,29,39,38)(32,40,37,34);(2,5)(3,7)(9,12,14,21)(10,16,13,20)(11,18,19,22)(15,23,17,24)(25,28)(26,33)(27,38)(29,39)(30,36)(31,35)(32,34)(37,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,24,22,23)(25,29,31,38)(26,34,36,40)(27,28,39,35)(30,37,33,32);(1,5,6,2)(3,8,7,4)(9,11)(10,17)(12,22)(13,15)(14,19)(16,23)(18,21)(20,24)(25,30,31,33)(26,35,36,28)(27,34,39,40)(29,32,38,37)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=ghi,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, 8627555);