#7641945

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	#37515	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4478 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)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(26,30)(27,33)(32,37)(34,39);(1,2)(3,7)(4,8)(5,6)(10,15)(12,16)(18,21)(20,23)(28,35)(32,37)(34,39)(38,40);(1,3,5,4)(2,7,6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,5,4)(2,7,6,8)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,3,5,4)(2,7,6,8)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40)

Polycyclic presentation:

2^9|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, 7549141);