#8917763

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

4474 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8824959);