#7642749

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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)(12,19)(16,21)(17,23)(22,24)(26,30)(27,33)(32,35)(34,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24);(1,3,5,4)(2,7,6,8)(10,14)(15,20)(16,21)(22,24)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(1,3,5,4)(2,7,6,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39)

Polycyclic presentation:

2^9|c^2=gh,d^2=gi,e^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, 7549945);