#10220712

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

4195 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,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, 10127908);