#7652044

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

4560 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,e^2=ghi,f^2=gh,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, 7559240);