#10325524

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

4353 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:

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

Polycyclic presentation:

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

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, 10232720);