#9053929

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	#52633	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(3,7)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,18)(19,24)(20,30)(21,25)(22,31)(23,26)(27,28)(29,32)(34,38)(35,43)(36,46)(42,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19,23,29)(18,24,26,32)(20,25,31,27)(21,22,28,30)(33,34,39,38)(35,44,43,40)(36,41,46,47)(37,48,45,42);(1,2,5,9)(3,6,11,7)(4,12,13,8)(10,15,16,14)(17,20,23,31)(18,22,26,30)(19,27,29,25)(21,32,28,24)(33,35,39,43)(34,40,38,44)(36,45,46,37)(41,48,47,42);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,20)(18,22)(19,25)(21,32)(23,31)(24,28)(26,30)(27,29)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,21,23,28)(18,25,26,27)(19,22,29,30)(20,24,31,32)(33,37,39,45)(34,42,38,48)(35,36,43,46)(40,41,44,47);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,22)(18,20)(19,28)(21,29)(23,30)(24,25)(26,31)(27,32)(33,38,39,34)(35,44,43,40)(36,47,46,41)(37,48,45,42)

Polycyclic presentation:

2^9|b^2=i,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961125);