#10103344

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	#38516	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4218 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=g,e^2=h,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, 10010540);