#8944515

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

4613 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^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, 8851711);