#10143317

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	#47911	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,23,25)(19,24,29,31)(20,26,28,22)(21,27,30,32)(33,34)(35,45)(36,42)(37,38)(39,43)(40,44)(41,48)(46,47);(2,7)(4,15)(8,11)(10,13)(17,19)(18,24)(20,30)(21,28)(22,27)(23,29)(25,31)(26,32)(33,35,40,48)(34,41,44,45)(36,46,37,39)(38,47,42,43);(2,7)(3,12)(9,14)(10,13)(18,25)(19,29)(20,28)(27,32)(33,36,40,37)(34,42,44,38)(35,46,48,39)(41,43,45,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,23,28)(18,26,25,22)(19,30,29,21)(24,32,31,27)(33,37,40,36)(34,42,44,38)(35,39,48,46)(41,43,45,47);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,21,23,30)(18,27,25,32)(19,20,29,28)(22,24,26,31)(33,38,40,42)(34,37,44,36)(35,47,48,43)(39,45,46,41);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,26)(18,28,25,20)(19,27,29,32)(21,31,30,24)(33,39)(34,43)(35,36)(37,48)(38,41)(40,46)(42,45)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10050513);