#10102642

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

4455 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,e^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, 10009838);