#10040586

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3799 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9947782);