#9136906

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	#38849	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=h,e^2=gh,f^2=gh,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, 9044102);