#10142170

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

4258 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,22)(19,25,20,24)(21,26,29,30)(27,32,28,31)(33,34,38,39)(35,36,41,40)(37,45,47,42)(43,46,48,44);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,20)(18,24,22,25)(21,27,29,28)(26,31,30,32)(34,39)(36,40)(42,45)(44,46);(2,3)(7,9)(8,10)(14,15)(33,35,38,41)(34,40,39,36)(37,43,47,48)(42,46,45,44);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,19)(18,25,22,24)(21,27,29,28)(26,31,30,32)(33,35)(34,40)(36,39)(37,43)(38,41)(42,46)(44,45)(47,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,36,38,40)(34,41,39,35)(37,46,47,44)(42,43,45,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,18)(19,25,20,24)(21,26,29,30)(27,31,28,32)(33,37,38,47)(34,42,39,45)(35,43,41,48)(36,44,40,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=ghi,e^2=h,f^2=g,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, 10049366);