#10195723

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

4274 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)(9,10,15,17)(11,16,19,23)(12,14,21,24)(13,18,20,22)(26,32)(28,38)(33,35)(36,39);(1,3)(2,6)(4,8)(5,7)(9,11,15,19)(10,16,17,23)(12,20,21,13)(14,22,24,18)(25,26,31,32)(27,33,29,35)(28,34,38,30)(36,40,39,37);(1,4)(2,7)(3,8)(5,6)(10,17)(11,19)(12,21)(18,22)(25,27,31,29)(26,33,32,35)(28,36,38,39)(30,37,34,40);(2,8)(4,6)(9,12)(10,14)(11,20)(13,19)(15,21)(16,22)(17,24)(18,23)(25,28)(26,34)(27,36)(29,39)(30,32)(31,38)(33,40)(35,37);(1,4)(2,3)(5,6)(7,8)(9,13,15,20)(10,18,17,22)(11,21,19,12)(14,16,24,23)(25,29,31,27)(26,35,32,33)(28,36,38,39)(30,37,34,40);(1,3)(2,6)(4,8)(5,7)(9,14)(10,12)(11,22)(13,23)(15,24)(16,20)(17,21)(18,19)(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=ghi,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, 10102919);