#10144299

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

4364 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10051495);