#10144545

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

4211 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)(11,20)(12,22)(13,19)(14,16)(15,17)(18,23)(21,24)(25,26,32,30)(27,36,38,33)(28,34,29,39)(31,35,37,40);(1,2)(3,4)(5,7)(6,8)(9,11,15,14)(10,16,17,20)(12,21,23,19)(13,18,24,22)(25,27)(26,33)(28,31)(29,37)(30,36)(32,38)(34,40)(35,39);(1,3)(2,6)(4,7)(5,8)(10,17)(12,23)(16,20)(19,21)(25,28,32,29)(26,34,30,39)(27,31,38,37)(33,40,36,35);(1,3)(2,4)(5,8)(6,7)(9,12)(10,18)(11,21)(13,20)(14,19)(15,23)(16,24)(17,22)(25,29,32,28)(26,34,30,39)(27,37,38,31)(33,40,36,35);(1,4,5,6)(2,3,7,8)(9,13,15,24)(10,19,17,21)(11,22,14,18)(12,16,23,20)(25,30,32,26)(27,36,38,33)(28,39,29,34)(31,35,37,40);(1,3)(2,4)(5,8)(6,7)(9,14,15,11)(10,20,17,16)(12,19,23,21)(13,22,24,18)(25,31,32,37)(26,35,30,40)(27,28,38,29)(33,39,36,34)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10051741);