#9434721

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	#37805	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,f^2=g,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, 9341917);