#10455748

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

4595 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)(9,10,13,16)(11,17,18,14)(12,15,22,21)(19,24,20,23)(25,26,32,30)(27,29,33,28)(31,38,39,34)(35,36,40,37);(11,18)(14,17)(19,20)(23,24)(25,27)(26,28)(29,30)(31,35)(32,33)(34,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,17)(20,22)(21,24)(25,28,32,29)(26,33,30,27)(31,36,39,37)(34,40,38,35);(3,7)(6,8)(9,11)(10,14)(12,20)(13,18)(15,24)(16,17)(19,22)(21,23)(25,29,32,28)(26,33,30,27)(31,37,39,36)(34,40,38,35);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,19)(13,22)(14,23)(16,21)(17,24)(18,20)(25,30,32,26)(27,29,33,28)(31,34,39,38)(35,36,40,37);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,21)(13,16)(15,22)(17,18)(19,24)(20,23)(25,31)(26,34)(27,35)(28,36)(29,37)(30,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=g,e^2=g,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, 10362944);