#9005275

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	#53399	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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)(3,7,5,8)(25,26)(27,33)(28,35)(29,37)(30,32)(31,38)(34,36)(39,40);(1,3,6,5)(2,7,4,8)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,3,6,5)(2,8,4,7)(10,12)(11,17)(13,21)(22,24)(25,27)(26,28)(29,34)(30,35)(31,36)(32,33)(37,39)(38,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,4,6,2)(3,7,5,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,38)(39,40);(1,5,6,3)(2,7,4,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8912471);