#9053862

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	#35730	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4344 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)(9,10)(11,12)(13,15)(14,16)(17,19)(18,20)(21,22)(23,24)(26,32)(27,37)(28,30)(35,40);(3,7)(6,8)(9,10,14,16)(11,12,17,19)(13,15,21,22)(18,20,23,24)(25,26)(27,38)(28,34)(29,40)(30,36)(31,32)(33,37)(35,39);(1,2)(3,6)(4,5)(7,8)(11,17)(12,19)(13,21)(15,22)(25,27)(26,33)(28,29)(30,39)(31,37)(32,38)(34,35)(36,40);(9,11,14,17)(10,12,16,19)(13,18,21,23)(15,20,22,24)(25,28,31,30)(26,34,32,36)(27,29,37,39)(33,35,38,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,29,31,39)(26,35,32,40)(27,28,37,30)(33,34,38,36);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,15,16,22)(11,18,17,23)(12,20,19,24)(25,30,31,28)(26,36,32,34)(27,39,37,29)(33,40,38,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961058);