#7714584

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621780);