#10143501

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=i,d^2=hi,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, 10050697);