#9729220

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	#53503	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,14)(13,15)(17,18)(19,20)(21,25)(22,24)(23,27)(26,28)(29,31)(30,32)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(1,3,6,5)(2,7,9,10)(4,11,13,12)(8,14,15,16)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(18,24)(20,23)(25,32)(28,31)(33,35,37,43)(34,39,41,38)(36,42,45,48)(40,47,44,46);(1,4,6,13)(2,8,9,15)(3,11,5,12)(7,14,10,16)(17,20,22,23)(18,19,24,27)(21,28,30,31)(25,26,32,29)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5,6,3)(2,7,9,10)(4,12,13,11)(8,14,15,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,34)(35,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,16)(13,15)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=ghi,f^2=i,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, 9636416);