#8922536

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

4552 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829732);