#9508819

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	#52048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3838 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)(33,34,39,43)(35,36,46,40)(37,41,45,48)(38,42,44,47);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(17,18,22,23)(19,21,25,30)(20,24,27,31)(26,29,28,32)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,42)(47,48);(2,3)(4,11)(5,13)(14,15)(18,23)(20,27)(21,30)(26,28)(37,45)(38,44)(41,48)(42,47);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19,22,25)(18,21,23,30)(20,28,27,26)(24,32,31,29)(33,36,39,40)(34,35,43,46)(37,42,45,47)(38,48,44,41);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37,39,45)(34,41,43,48)(35,44,46,38)(36,47,40,42);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21)(18,25)(19,23)(20,29)(22,30)(24,28)(26,31)(27,32)(33,38,39,44)(34,42,43,47)(35,45,46,37)(36,48,40,41)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=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, 9416015);