#10512752

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

4625 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10419948);