#9290455

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

4351 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9197651);