#9062093

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	#44779	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4491 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,e^2=g,f^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, 8969289);