#10220401

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

4200 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gh,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, 10127597);