#10143602

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	#47603	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=ghi,e^2=gh,f^2=h,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, 10050798);