#9390491

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

4040 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,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, 9297687);