#8935053

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

4527 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)(11,13)(12,21)(14,22)(15,17)(16,20)(18,19)(23,24)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,20)(10,16,17,13)(12,18,22,24)(14,19,21,23)(27,34)(30,31)(35,37)(39,40);(2,5)(3,7)(10,17)(13,16)(14,21)(19,23)(25,27,29,34)(26,30,33,31)(28,37,38,35)(32,40,36,39);(3,7)(6,8)(9,12)(10,14)(11,18)(13,23)(15,22)(16,19)(17,21)(20,24)(25,26)(27,30)(28,32)(29,33)(31,34)(35,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,16)(10,11,17,20)(12,19,22,23)(14,24,21,18)(25,27)(26,31)(28,35)(29,34)(30,33)(32,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,14)(10,12)(11,19)(13,24)(15,21)(16,18)(17,22)(20,23)(25,28)(26,32)(27,35)(29,38)(30,39)(31,40)(33,36)(34,37)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,e^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, 8842249);