#10411877

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

4524 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,e^2=g,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, 10319073);