#7641938

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

4454 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)(11,12)(13,21)(14,15)(16,19)(17,22)(18,23)(20,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,2)(3,7)(4,8)(5,6)(10,15)(12,16)(17,21)(20,23)(26,31)(27,33)(34,37)(35,40);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,29,30,38)(26,34,31,37)(27,35,33,40)(28,36,32,39)

Polycyclic presentation:

2^9|c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=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, 7549134);