#9062156

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	#44878	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^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, 8969352);