#8973329

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

4374 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)(4,5)(6,8)(9,10,14,16)(11,17,18,12)(13,15,21,22)(19,23,24,20)(25,26,30,31)(27,32,35,39)(28,33,37,36)(29,38,40,34);(2,5)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,26,30,31)(27,32,35,39)(28,36,37,33)(29,34,40,38);(1,2)(3,6)(4,5)(7,8)(11,18)(12,17)(13,21)(15,22)(26,31)(27,35)(28,37)(29,40);(1,3,4,8)(2,6,5,7)(9,11,14,18)(10,12,16,17)(13,19,21,24)(15,20,22,23)(25,27)(26,32)(28,38)(29,36)(30,35)(31,39)(33,40)(34,37);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,28,30,37)(26,33,31,36)(27,34,35,38)(29,39,40,32);(9,13)(10,15)(11,19)(12,20)(14,21)(16,22)(17,23)(18,24)(25,29)(26,34)(27,36)(28,39)(30,40)(31,38)(32,37)(33,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=g,e^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, 8880525);