#9058638

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
prime radical	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4429 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965834);