#9546039

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,23,26)(19,24,21,22)(20,25,29,32)(27,31,30,28)(33,34)(35,41)(36,46)(37,43)(38,44)(39,40)(42,47)(45,48);(2,7)(4,15)(8,11)(10,13)(17,19,23,21)(18,24,26,22)(20,27,29,30)(25,31,32,28)(33,35,40,37)(34,41,39,43)(36,38,47,45)(42,48,46,44);(2,7)(3,12)(4,15)(5,16)(19,21)(22,24)(27,30)(28,31)(33,36,40,47)(34,42,39,46)(35,45,37,38)(41,44,43,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,23,29)(18,25,26,32)(19,27,21,30)(22,28,24,31)(33,37,40,35)(34,43,39,41)(36,45,47,38)(42,44,46,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21,23,19)(18,22,26,24)(20,27,29,30)(25,31,32,28)(33,38,40,45)(34,44,39,48)(35,36,37,47)(41,46,43,42);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22)(18,19)(20,28)(21,26)(23,24)(25,27)(29,31)(30,32)(33,39,40,34)(35,43,37,41)(36,42,47,46)(38,48,45,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gh,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,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, 9453235);