#9029122

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	#49584	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=h,e^2=g,f^2=i,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, 8936318);