#9143365

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	#48461	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4249 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9050561);