#10411766

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

4707 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)(3,4)(5,6)(7,8)(9,10,13,14)(11,17,19,15)(12,20,22,16)(18,23,21,24)(25,26,31,34)(27,32,28,29)(30,33,38,40)(35,39,36,37);(1,3,7,6)(2,4,8,5)(10,14)(15,17)(16,20)(23,24)(25,27,31,28)(26,29,34,32)(30,35,38,36)(33,37,40,39);(1,4,7,5)(2,6,8,3)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,24)(21,22)(25,27,31,28)(26,32,34,29)(30,35,38,36)(33,39,40,37);(1,4,7,5)(2,3,8,6)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,28,31,27)(26,32,34,29)(30,35,38,36)(33,37,40,39);(1,5,7,4)(2,3,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,29,31,32)(26,28,34,27)(30,37,38,39)(33,36,40,35);(1,6,7,3)(2,4,8,5)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10318962);