#9778473

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	#43892	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9685669);