#9712322

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
prime radical	#57971	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4346 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)(4,9)(5,10)(7,8)(11,15)(12,14)(13,16)(17,18)(19,27)(20,23)(21,25)(22,30)(24,28)(26,31)(29,32)(34,38)(36,44)(39,41)(42,48);(1,3)(2,6)(4,5)(7,11)(8,15)(9,10)(12,13)(14,16)(17,19,23,28)(18,24,20,27)(21,31,32,22)(25,30,29,26)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,8)(3,11)(4,12)(10,16)(18,20)(24,27)(25,29)(26,30)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,4,7,12)(2,9,8,14)(3,5,11,13)(6,10,15,16)(17,20,23,18)(19,27,28,24)(21,29,32,25)(22,30,31,26)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5)(2,10)(3,12)(4,11)(6,14)(7,13)(8,16)(9,15)(17,21,23,32)(18,25,20,29)(19,22,28,31)(24,26,27,30)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,6,7,15)(2,11,8,3)(4,10,12,16)(5,9,13,14)(17,22)(18,26)(19,21)(20,30)(23,31)(24,25)(27,29)(28,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,d^2=gi,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,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, 9619518);