#10247667

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

4533 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10154863);