#9027287

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

4089 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,15)(13,14)(17,18)(19,27)(20,25)(21,31)(22,23)(24,29)(26,32)(28,30)(33,34,38,41)(35,39,43,37)(36,40,46,48)(42,47,44,45);(1,3,7,6)(2,8,9,16)(4,12,14,11)(5,10,15,13)(18,23)(20,30)(24,27)(26,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,46)(45,48);(2,9)(3,6)(4,14)(5,15)(17,19,22,29)(18,24,23,27)(20,31,30,26)(21,25,32,28)(33,35,38,43)(34,39,41,37)(36,44,46,42)(40,45,48,47);(1,4)(2,10)(3,12)(5,16)(6,11)(7,14)(8,15)(9,13)(17,19)(18,24)(20,31)(21,28)(22,29)(23,27)(25,32)(26,30)(33,35)(34,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,5,7,15)(2,11,9,12)(3,13,6,10)(4,8,14,16)(17,20,22,30)(18,25,23,28)(19,26,29,31)(21,27,32,24)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,6,7,3)(2,8,9,16)(4,11,14,12)(5,10,15,13)(17,21)(18,26)(19,28)(20,27)(22,32)(23,31)(24,30)(25,29)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,48,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8934483);