#8359850

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3936 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)(3,9,10,5)(4,12,13,8)(11,15,14,16)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,34,39,36)(35,40,42,38)(37,41,47,44)(43,48,45,46);(2,7)(3,10)(4,13)(15,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,39,42)(34,40,36,38)(37,45,47,43)(41,46,44,48);(2,7)(5,9)(8,12)(15,16)(22,30)(26,29)(27,28)(31,32)(35,42)(38,40)(43,45)(46,48);(1,3)(2,5)(4,14)(6,10)(7,9)(8,16)(11,13)(12,15)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36,39,34)(35,38,42,40)(37,41,47,44)(43,48,45,46);(1,4,6,13)(2,8,7,12)(3,11,10,14)(5,15,9,16)(17,21,23,18)(19,24,20,25)(22,29,30,26)(27,31,28,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45);(1,5)(2,3)(4,15)(6,9)(7,10)(8,11)(12,14)(13,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38)(34,35)(36,42)(37,46)(39,40)(41,43)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8267046);