#9267023

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

3976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=i,e^2=gh,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, 9174219);