#7657457

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

4106 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=i,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7564653);