#10220393

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

4217 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=h,e^2=g,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, 10127589);