#9050471

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

4310 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)(17,18,23,21)(19,22,27,24)(20,25,28,30)(26,29,32,31)(33,34,38,36)(35,39,42,43)(37,40,46,44)(41,47,45,48);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,19,23,27)(18,24,21,22)(20,26,28,32)(25,31,30,29)(34,36)(39,43)(40,44)(47,48);(2,6)(3,9)(4,12)(14,16)(18,21)(20,28)(22,24)(26,32)(33,35,38,42)(34,39,36,43)(37,45,46,41)(40,48,44,47);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,25)(19,26)(21,30)(22,29)(23,28)(24,31)(27,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,47)(44,48)(45,46);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,21,23,18)(19,22,27,24)(20,25,28,30)(26,31,32,29)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,22)(18,19)(20,29)(21,27)(23,24)(25,26)(28,31)(30,32)(33,37)(34,40)(35,41)(36,44)(38,46)(39,47)(42,45)(43,48)

Polycyclic presentation:

2^9|a^2=g,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, 8957667);