#8818180

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

4035 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725376);