#10201919

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

4302 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)(3,4)(5,8)(6,7)(9,10,15,19)(11,16,13,18)(12,22,14,17)(20,24,21,23)(26,32)(28,38)(33,35)(36,39);(1,3)(2,6)(4,8)(5,7)(9,11,15,13)(10,16,19,18)(12,21,14,20)(17,24,22,23)(25,26,31,32)(27,33,29,35)(28,34,38,30)(36,40,39,37);(1,4)(2,7)(3,8)(5,6)(9,12,15,14)(10,17,19,22)(11,20,13,21)(16,23,18,24)(25,27,31,29)(26,33,32,35)(28,36,38,39)(30,37,34,40);(2,8)(4,6)(9,13,15,11)(10,18,19,16)(12,20,14,21)(17,23,22,24)(25,28)(26,34)(27,36)(29,39)(30,32)(31,38)(33,40)(35,37);(1,4)(2,3)(5,6)(7,8)(9,14,15,12)(10,17,19,22)(11,21,13,20)(16,23,18,24)(25,29,31,27)(26,35,32,33)(28,36,38,39)(30,37,34,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,16)(12,17)(13,18)(14,22)(15,19)(20,23)(21,24)(25,30,31,34)(26,28,32,38)(27,37,29,40)(33,36,35,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=i,e^2=gh,f^2=ghi,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, 10109115);