#8993773

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

4370 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)(3,4)(5,6)(7,8)(9,10,14,16)(11,15,19,22)(12,17,21,24)(13,20,23,18)(25,26,30,32)(27,33,34,28)(29,31,37,38)(35,39,40,36);(2,6)(3,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,21)(23,24)(25,26)(27,28)(29,31)(30,32)(33,34)(35,36)(37,38)(39,40);(1,3,5,7)(2,4,6,8)(10,16)(11,19)(12,21)(13,23)(27,34)(28,33)(29,37)(31,38);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,21)(23,24)(25,27,30,34)(26,28,32,33)(29,35,37,40)(31,36,38,39);(9,12)(10,17)(11,20)(13,22)(14,21)(15,23)(16,24)(18,19)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(9,13,14,23)(10,18,16,20)(11,17,19,24)(12,22,21,15)(25,29)(26,31)(27,35)(28,36)(30,37)(32,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=ghi,f^2=h,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, 8900969);