#8824144

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

4296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=gi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731340);