#7641921

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	#35130	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4426 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=g,e^2=h,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, 7549117);