#7646777

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

4112 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)(10,15)(11,16)(13,21)(19,24)(26,30)(27,33)(32,37)(34,39);(1,3,4,8)(2,5,6,7)(26,30)(31,36)(32,37)(38,40);(9,10,14,15)(11,12,16,20)(13,17,21,22)(18,19,23,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,23)(22,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=i,e^2=i,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, 7553973);