#9020227

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	#44026	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8927423);