#8899452

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

4472 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806648);