#9668755

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

4560 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)(11,20)(12,13)(14,21)(15,17)(16,22)(18,23)(19,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,22)(10,16,17,20)(12,21,23,19)(13,24,18,14)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,3)(2,5)(4,8)(6,7)(10,17)(12,23)(16,20)(19,21)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(9,12)(10,18)(11,21)(13,17)(14,16)(15,23)(19,22)(20,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13,15,18)(10,12,17,23)(11,14,22,24)(16,19,20,21)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,2)(3,5)(4,7)(6,8)(9,14)(10,19)(11,13)(12,16)(15,24)(17,21)(18,22)(20,23)(25,28)(26,31)(27,33)(29,36)(30,37)(32,35)(34,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575951);