#7643510

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	#35301	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4532 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,e^2=h,f^2=hi,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, 7550706);