#7641923

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	C₂²×#41	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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=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, 7549119);