#8946243

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

4142 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)(3,9)(4,8)(5,11)(6,7)(10,16)(12,14)(13,15)(17,18,23,21)(19,24,22,29)(20,25,28,30)(26,31,27,32)(33,34,39,43)(35,42,47,40)(36,41,37,38)(44,46,45,48);(1,3)(2,5)(4,12)(6,11)(7,9)(8,16)(10,13)(14,15)(17,19,23,22)(18,24,21,29)(20,26,28,27)(25,31,30,32)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(2,7)(4,13)(5,9)(10,12)(18,21)(19,22)(25,30)(26,27)(33,36,39,37)(34,41,43,38)(35,44,47,45)(40,48,42,46);(1,3)(2,5)(4,12)(6,11)(7,9)(8,16)(10,13)(14,15)(17,20,23,28)(18,25,21,30)(19,26,22,27)(24,31,29,32)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,4,6,13)(2,8,7,15)(3,10,11,12)(5,14,9,16)(17,21,23,18)(19,24,22,29)(20,25,28,30)(26,32,27,31)(33,35)(34,42)(36,45)(37,44)(38,46)(39,47)(40,43)(41,48);(1,5)(2,3)(4,14)(6,9)(7,11)(8,10)(12,15)(13,16)(17,22,23,19)(18,24,21,29)(20,27,28,26)(25,31,30,32)(33,38,39,41)(34,36,43,37)(35,46,47,48)(40,44,42,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8853439);