#10142169

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

4054 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 10049365);