#9050470

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

4146 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8957666);