#7651868

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

4481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7559064);