#10143381

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	#50677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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)(3,6)(4,9)(5,10)(7,8)(11,16)(12,14)(13,15)(17,18,22,24)(19,20,27,23)(21,25,31,32)(26,28,29,30)(33,34)(35,43)(36,41)(37,46)(38,40)(39,42)(44,48)(45,47);(1,3)(2,6)(4,12)(5,13)(7,11)(8,16)(9,14)(10,15)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,35,39,38)(34,40,42,43)(36,44,37,45)(41,47,46,48);(2,8)(3,11)(4,13)(10,14)(18,24)(20,23)(25,32)(28,30)(33,36,39,37)(34,41,42,46)(35,44,38,45)(40,47,43,48);(1,4,7,13)(2,9,8,15)(3,12,11,5)(6,14,16,10)(17,20,22,23)(18,19,24,27)(21,30,31,28)(25,29,32,26)(33,37,39,36)(34,41,42,46)(35,45,38,44)(40,47,43,48);(1,5,7,12)(2,10,8,14)(3,4,11,13)(6,9,16,15)(17,21,22,31)(18,25,24,32)(19,26,27,29)(20,28,23,30)(33,34)(35,40)(36,41)(37,46)(38,43)(39,42)(44,47)(45,48);(1,6,7,16)(2,11,8,3)(4,14,13,10)(5,15,12,9)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,38,39,35)(34,40,42,43)(36,45,37,44)(41,47,46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=gi,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, 10050577);