#9027054

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

4247 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=gh,e^2=g,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, 8934250);