#10143603

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

4212 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)(17,18,23,25)(19,24,28,22)(20,26,30,31)(21,27,29,32)(33,34,38,39)(35,43,44,40)(36,41,45,48)(37,46,47,42);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,28)(18,24,25,22)(20,29,30,21)(26,32,31,27)(34,39)(36,45)(40,43)(42,46);(2,3)(7,9)(8,10)(14,15)(18,25)(19,28)(20,30)(27,32)(33,35,38,44)(34,40,39,43)(36,42,45,46)(37,41,47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,26)(19,29)(21,28)(22,27)(23,30)(24,32)(25,31)(33,35)(34,40)(36,46)(37,41)(38,44)(39,43)(42,45)(47,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,29)(18,27,25,32)(19,30,28,20)(22,26,24,31)(33,36)(34,41)(35,42)(37,43)(38,45)(39,48)(40,47)(44,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,24)(18,28,25,19)(20,27,30,32)(21,31,29,26)(33,37,38,47)(34,42,39,46)(35,41,44,48)(36,40,45,43)

Polycyclic presentation:

2^9|a^2=gi,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, 10050799);