#9390487

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

4213 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297683);