#8899543

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

4520 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)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(25,26)(27,34)(28,38)(29,39)(30,37)(31,35)(32,40)(33,36);(3,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,27,31,37)(26,30,35,34)(28,36,40,29)(32,33,38,39);(1,2)(3,6)(4,5)(7,8)(11,17)(15,19)(18,23)(22,24)(25,28,31,40)(26,32,35,38)(27,36,37,29)(30,33,34,39);(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29)(26,33)(27,28)(30,38)(31,36)(32,34)(35,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,10)(11,19,17,15)(12,16,20,21)(18,22,23,24)(25,27)(26,34)(28,36)(29,40)(30,35)(31,37)(32,39)(33,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,30,31,34)(26,27,35,37)(28,39,40,33)(29,38,36,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806739);