#9236446

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 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)(3,8)(4,15)(5,10)(6,14)(7,11)(9,13)(12,16)(17,18)(19,29)(20,25)(21,26)(22,32)(23,28)(24,30)(27,31)(33,34,40,39)(35,44,38,41)(36,42,47,45)(37,43,48,46);(1,3)(2,8)(4,5)(6,11)(7,14)(9,16)(10,15)(12,13)(17,19,23,30)(18,24,28,29)(20,27,26,32)(21,31,25,22)(33,35,40,38)(34,41,39,44)(36,43,47,46)(37,45,48,42);(1,4)(2,9)(3,12)(5,14)(6,16)(7,13)(8,10)(11,15)(17,20)(18,21)(19,27)(22,29)(23,26)(24,31)(25,28)(30,32)(33,36,40,47)(34,42,39,45)(35,43,38,46)(37,44,48,41);(1,4)(2,9)(3,5)(6,10)(7,13)(8,16)(11,15)(12,14)(17,20)(18,25)(19,27)(21,28)(22,24)(23,26)(29,31)(30,32)(33,37)(34,43)(35,45)(36,44)(38,42)(39,46)(40,48)(41,47);(1,5)(2,10)(3,13)(4,14)(6,15)(7,12)(8,9)(11,16)(17,21,23,25)(18,26,28,20)(19,22,30,31)(24,27,29,32)(33,38,40,35)(34,41,39,44)(36,46,47,43)(37,45,48,42);(1,6,7,8)(2,3,11,14)(4,16,13,10)(5,9,12,15)(17,22)(18,27)(19,21)(20,29)(23,31)(24,26)(25,30)(28,32)(33,39,40,34)(35,44,38,41)(36,45,47,42)(37,43,48,46)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9143642);