#9057775

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	#46335	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4346 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=g,f^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, 8964971);