#7709312

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

4352 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7616508);