#10512757

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

4641 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)(9,10,14,17)(11,15,19,12)(13,16,22,24)(18,21,23,20)(25,26,31,35)(27,32,29,39)(28,33,34,30)(36,38,37,40);(9,11,14,19)(10,12,17,15)(13,18,22,23)(16,20,24,21)(25,27)(26,32)(28,37)(29,31)(30,38)(33,40)(34,36)(35,39);(1,3,6,5)(2,7,4,8)(25,28)(26,30)(27,36)(29,37)(31,34)(32,40)(33,35)(38,39);(1,3,6,5)(2,8,4,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23)(25,29,31,27)(26,32,35,39)(28,37,34,36)(30,40,33,38);(1,4,6,2)(3,8,5,7)(9,12,14,15)(10,11,17,19)(13,21,22,20)(16,23,24,18)(25,28)(26,33)(27,37)(29,36)(30,35)(31,34)(32,40)(38,39);(1,5,6,3)(2,8,4,7)(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,24)(19,23)(25,30,31,33)(26,34,35,28)(27,38,29,40)(32,36,39,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=h,e^2=ghi,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, 10419953);