#8873738

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

4345 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780934);