#9058491

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

4140 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965687);