#9017452

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4512 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26)(27,36)(28,39)(29,30)(31,32)(33,37)(34,38)(35,40);(3,8)(6,7)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(26,32)(27,37)(29,35)(34,39);(1,3,5,8)(2,6,4,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27)(26,33)(28,35)(29,38)(30,39)(31,37)(32,36)(34,40);(3,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,28,31,38)(26,34,32,39)(27,35,37,29)(30,33,40,36);(1,3)(2,7)(4,6)(5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,24)(20,22)(21,23)(25,29,31,35)(26,30,32,40)(27,38,37,28)(33,39,36,34);(1,4,5,2)(3,6,8,7)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,30)(26,35)(27,39)(28,33)(29,32)(31,40)(34,37)(36,38)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=ghi,e^2=ghi,f^2=gh,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, 8924648);