#9669375

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

4477 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=g,e^2=gh,f^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, 9576571);