#9729206

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

4121 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,29)(30,32)(35,42)(37,39)(43,48)(45,47);(2,7)(4,8)(10,13)(12,14)(17,19,21,26)(18,22,23,29)(20,25,28,32)(24,30,27,31)(33,34)(35,37)(36,44)(38,41)(39,42)(40,46)(43,45)(47,48);(2,7)(3,9)(10,13)(11,16)(18,23)(22,29)(24,27)(30,31)(33,35,38,42)(34,39,41,37)(36,43,46,48)(40,45,44,47);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,28)(31,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,47)(44,45)(46,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,36,38,46)(34,40,41,44)(35,43,42,48)(37,47,39,45);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,20,21,28)(18,24,23,27)(19,25,26,32)(22,30,29,31)(33,37)(34,35)(36,45)(38,39)(40,48)(41,42)(43,44)(46,47)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=gi,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, 9636402);