#9668293

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

3875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 9575489);