#7645818

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

4370 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)(10,16)(14,18)(17,19)(21,23)(25,26)(27,34)(28,37)(29,32)(30,38)(31,33)(35,36)(39,40);(1,2)(3,7)(4,8)(5,6)(9,10,15,16)(11,19,12,17)(13,18,24,14)(20,21,22,23);(9,11,15,12)(10,17,16,19)(13,20,24,22)(14,21,18,23)(25,27,31,29)(26,32,33,34)(28,35,38,39)(30,36,37,40);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,17,16,19)(13,20,24,22)(14,23,18,21)(25,28)(26,30)(27,35)(29,39)(31,38)(32,36)(33,37)(34,40);(9,13)(10,18)(11,20)(12,22)(14,16)(15,24)(17,23)(19,21)(25,29,31,27)(26,32,33,34)(28,35,38,39)(30,40,37,36);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,13,16,24)(11,21,12,23)(17,20,19,22)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,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^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553014);