#10188287

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

4427 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,e^2=g,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, 10095483);