#10143417

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	#51286	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=gh,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, 10050613);