#10320384

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) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,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, 10227580);