#7642567

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	#39160	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4390 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)(10,15)(13,17)(16,22)(19,23)(26,30)(27,33)(32,35)(34,39);(1,2)(3,7)(4,8)(5,6)(9,10,14,15)(11,16,20,22)(12,17,21,13)(18,23,24,19);(10,15)(12,21)(16,22)(18,24)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(1,3,5,4)(2,7,6,8)(9,11,14,20)(10,16,15,22)(12,18,21,24)(13,19,17,23)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39);(9,12)(10,17)(11,18)(13,15)(14,21)(16,23)(19,22)(20,24)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40);(1,4,5,3)(2,8,6,7)(9,13,14,17)(10,12,15,21)(11,19,20,23)(16,18,22,24)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=g,e^2=h,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, 7549763);