#9729218

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,14)(12,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,30)(29,31)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(2,7)(4,12)(8,14)(10,13)(17,19,22,21)(18,23,25,26)(20,27,29,28)(24,30,31,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,7)(3,11)(9,16)(10,13)(33,35,37,43)(34,39,41,38)(36,42,45,48)(40,47,44,46);(1,3,6,11)(2,8,7,14)(4,10,12,13)(5,9,15,16)(17,20,22,29)(18,24,25,31)(19,27,21,28)(23,30,26,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,4)(2,9)(3,10)(5,14)(6,12)(7,16)(8,15)(11,13)(17,21,22,19)(18,23,25,26)(20,28,29,27)(24,30,31,32)(33,34)(35,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,5,6,15)(2,10,7,13)(3,9,11,16)(4,8,12,14)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,32)(29,31)(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=h,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, 9636414);