#7642603

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

4449 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:

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

Polycyclic presentation:

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

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, 7549799);