#8963605

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

4476 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)(10,11)(15,17)(16,18)(22,23);(1,2,4,6)(3,5,8,7)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,26,30,28)(27,31,34,35)(29,32,37,36)(33,38,39,40);(1,2,4,6)(3,7,8,5)(12,19)(13,21)(15,17)(16,18)(26,28)(29,37)(31,35)(33,39);(1,2,4,6)(3,5,8,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27,30,34)(26,31,28,35)(29,33,37,39)(32,38,36,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,26)(27,31,34,35)(29,36,37,32)(33,38,39,40);(1,3,4,8)(2,5,6,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29)(26,32)(27,33)(28,36)(30,37)(31,38)(34,39)(35,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870801);