#9726510

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

4097 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,32)(30,31)(33,34)(35,36)(37,45)(38,39)(40,43)(41,47)(42,46)(44,48);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,22,29)(18,23,26,27)(20,21,28,31)(24,30,32,25)(34,39)(36,40)(41,45)(44,46);(2,7)(3,10)(8,15)(11,12)(33,35,38,43)(34,40,39,36)(37,42,47,48)(41,46,45,44);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,22,28)(18,24,26,32)(19,21,29,31)(23,30,27,25)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,21)(18,25)(19,28)(20,29)(22,31)(23,24)(26,30)(27,32)(33,36)(34,35)(37,46)(38,40)(39,43)(41,48)(42,45)(44,47);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,29)(28,32)(33,37,38,47)(34,41,39,45)(35,42,43,48)(36,44,40,46)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gh,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9633706);