#10143419

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	#56532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=hi,e^2=h,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^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10050615);