#9061245

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4390 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,e^2=h,f^2=g,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, 8968441);