#7642513

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

4359 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

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, 7549709);