#7709391

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

4366 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7616587);