#8983837

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

4571 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,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, 8891033);