#10143651

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	#57050	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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)(3,8)(4,10)(5,11)(6,14)(7,9)(12,15)(13,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34)(35,42)(36,40)(37,45)(38,39)(41,47)(43,44)(46,48);(1,3,7,12)(2,8,9,15)(4,13,14,5)(6,11,10,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(34,39)(36,42)(41,45)(44,46);(2,9)(3,12)(4,14)(11,16)(33,35,38,40)(34,36,39,42)(37,43,47,48)(41,44,45,46);(1,4)(2,10)(3,13)(5,12)(6,9)(7,14)(8,16)(11,15)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,46)(47,48);(1,5,7,13)(2,11,9,16)(3,14,12,4)(6,15,10,8)(17,21,22,18)(19,25,27,23)(20,24,29,28)(26,30,32,31)(33,36)(34,40)(35,39)(37,46)(38,42)(41,43)(44,47)(45,48);(1,6,7,10)(2,4,9,14)(3,11,12,16)(5,8,13,15)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,37,38,47)(34,41,39,45)(35,43,40,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,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, 10050847);