#10250323

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

4635 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=gi,e^2=g,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, 10157519);