#10158219

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

4721 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)(11,16)(12,13)(14,21)(15,17)(18,20)(19,23)(22,24)(25,26)(27,35)(28,37)(29,34)(30,38)(31,33)(32,36)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,20)(13,18)(14,22)(15,19)(17,23)(21,24)(25,27,31,36)(26,32,33,35)(28,34,38,40)(29,37,39,30);(1,3,5,4)(2,6,7,8)(10,17)(11,19)(12,22)(18,21)(26,33)(28,38)(32,35)(34,40);(1,2)(3,6)(4,8)(5,7)(9,12,15,22)(10,13,17,24)(11,20,19,14)(16,18,23,21)(25,28)(26,30)(27,34)(29,32)(31,38)(33,37)(35,39)(36,40);(1,4,5,3)(2,6,7,8)(9,13)(10,12)(11,21)(14,16)(15,24)(17,22)(18,19)(20,23)(25,29,31,39)(26,34,33,40)(27,30,36,37)(28,35,38,32);(1,2)(3,6)(4,8)(5,7)(9,14,15,20)(10,18,17,21)(11,22,19,12)(13,23,24,16)(25,30)(26,28)(27,29)(31,37)(32,34)(33,38)(35,40)(36,39)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=gh,e^2=hi,f^2=gh,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, 10065415);