#9676628

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×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4214 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=h,f^2=gh,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, 9583824);