#8107639

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4329 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014835);