#9143372

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	#40446	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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)(3,8)(4,12)(5,9)(6,10)(7,13)(11,15)(14,16)(17,18,23,20)(19,24,27,28)(21,25,31,30)(22,29,32,26)(33,34,39,44)(35,43,38,40)(36,46,47,41)(37,42,48,45);(1,3)(2,5)(4,11)(6,9)(7,15)(8,10)(12,16)(13,14)(17,19)(18,24)(20,28)(21,26)(22,25)(23,27)(29,31)(30,32)(33,35,39,38)(34,40,44,43)(36,45,47,42)(37,46,48,41);(4,13)(7,12)(11,14)(15,16)(18,20)(19,27)(25,30)(26,29)(33,36,39,47)(34,41,44,46)(35,45,38,42)(37,43,48,40);(1,4,6,13)(2,7,10,12)(3,11,9,14)(5,15,8,16)(17,20,23,18)(19,28,27,24)(21,25,31,30)(22,29,32,26)(33,36)(34,41)(35,45)(37,40)(38,42)(39,47)(43,48)(44,46);(1,3)(2,8)(4,14)(5,10)(6,9)(7,15)(11,13)(12,16)(17,21)(18,25)(19,29)(20,30)(22,28)(23,31)(24,32)(26,27)(33,37)(34,42)(35,41)(36,43)(38,46)(39,48)(40,47)(44,45);(1,5,6,8)(2,9,10,3)(4,15,13,16)(7,14,12,11)(17,22)(18,26)(19,25)(20,29)(21,24)(23,32)(27,30)(28,31)(33,38,39,35)(34,43,44,40)(36,42,47,45)(37,41,48,46)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=gh,f^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, 9050568);