#7641946

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	#37513	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4486 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)(11,13)(12,16)(14,22)(15,18)(17,23)(19,24)(20,21)(26,30)(27,33)(32,37)(34,39);(1,2,5,6)(3,7,4,8)(9,11,15,12)(10,16,18,13)(14,19,23,20)(17,24,22,21)(28,35)(32,37)(34,39)(38,40);(1,3,5,4)(2,7,6,8)(9,10)(11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,16,18,13)(14,20,23,19)(17,24,22,21)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,3,5,4)(2,7,6,8)(9,13,15,16)(10,11,18,12)(14,21,23,24)(17,19,22,20)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(1,4,5,3)(2,8,6,7)(9,14,15,23)(10,17,18,22)(11,19,12,20)(13,21,16,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,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^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, 7549142);