#10143705

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	#48619	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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=ghi,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, 10050901);