#10159317

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4141 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=g,e^2=ghi,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, 10066513);