#8901728

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

4364 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=gi,e^2=gi,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, 8808924);