#7655510

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

4442 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)(25,26,31,29)(27,34,32,28)(30,33,38,37)(35,40,39,36);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,15)(13,17)(14,19)(18,22)(20,23)(21,24);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(9,12,13,20)(10,15,17,23)(11,18,19,24)(14,21,16,22)(25,28,31,34)(26,32,29,27)(30,36,38,40)(33,39,37,35);(9,11)(10,16)(12,18)(13,19)(14,17)(15,22)(20,24)(21,23)(25,29,31,26)(27,34,32,28)(30,33,38,37)(35,36,39,40);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,23)(22,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7562706);