#8837167

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	#53174	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 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)(11,15)(12,16)(13,22)(14,18)(17,23)(19,20)(21,24)(25,26,31,30)(27,34,37,32)(28,29,39,33)(35,36,38,40);(3,7)(5,8)(9,11)(10,15)(12,13)(14,19)(16,22)(17,24)(18,20)(21,23)(25,27,31,37)(26,32,30,34)(28,38,39,35)(29,36,33,40);(1,2)(3,5)(4,6)(7,8)(11,19)(12,21)(15,20)(16,24)(25,28,31,39)(26,33,30,29)(27,35,37,38)(32,36,34,40);(9,12,14,21)(10,16,18,24)(11,13,19,23)(15,22,20,17)(25,29)(26,28)(27,36)(30,39)(31,33)(32,38)(34,35)(37,40);(1,3)(2,5)(4,7)(6,8)(9,13)(10,17)(11,12)(14,23)(15,24)(16,20)(18,22)(19,21)(25,30,31,26)(27,32,37,34)(28,29,39,33)(35,36,38,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,20)(12,16)(13,17)(14,18)(15,19)(21,24)(22,23)(25,29)(26,28)(27,36)(30,39)(31,33)(32,38)(34,35)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gi,e^2=g,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, 8744363);