#9434873

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4001 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9342069);