#9053855

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	#40554	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4176 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=gi,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, 8961051);