#8946539

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

4628 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)(11,17)(12,15)(13,21)(14,16)(18,22)(19,23)(20,24)(25,26,31,30)(27,33,28,32)(29,34,38,37)(35,40,36,39);(1,2)(3,4)(5,7)(6,8)(9,11,14,12)(10,15,16,17)(13,19,22,20)(18,23,21,24)(25,27,31,28)(26,32,30,33)(29,35,38,36)(34,39,37,40);(1,3)(2,6)(4,7)(5,8)(10,16)(13,22)(15,17)(19,20)(25,27,31,28)(26,33,30,32)(29,36,38,35)(34,39,37,40);(1,4,5,6)(2,3,7,8)(9,12,14,11)(10,17,16,15)(13,20,22,19)(18,24,21,23)(25,28,31,27)(26,33,30,32)(29,35,38,36)(34,39,37,40);(1,4)(2,3)(5,6)(7,8)(9,10)(11,17)(12,15)(13,18)(14,16)(19,24)(20,23)(21,22)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40);(1,2)(3,4)(5,7)(6,8)(9,13,14,22)(10,18,16,21)(11,19,12,20)(15,23,17,24)(25,30,31,26)(27,33,28,32)(29,37,38,34)(35,40,36,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=gi,f^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, 8853735);