#8992104

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

4303 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8899300);