#10143517

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	#48304	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(4,10)(5,11)(6,16)(7,9)(12,13)(14,15)(17,18,22,24)(19,20,27,23)(21,25,31,32)(26,28,29,30)(33,34,38,36)(35,39,43,40)(37,44,45,41)(42,47,48,46);(1,3,7,6)(2,8,9,16)(4,12,11,14)(5,15,10,13)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(2,9)(4,11)(8,16)(12,14)(18,24)(20,23)(25,32)(28,30)(33,35,38,43)(34,40,36,39)(37,42,45,48)(41,47,44,46);(1,4,7,11)(2,10,9,5)(3,12,6,14)(8,13,16,15)(17,20,22,23)(18,19,24,27)(21,30,31,28)(25,29,32,26)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(1,5,7,10)(2,11,9,4)(3,13,6,15)(8,12,16,14)(17,21,22,31)(18,25,24,32)(19,26,27,29)(20,28,23,30)(33,36,38,34)(35,39,43,40)(37,41,45,44)(42,47,48,46);(1,6,7,3)(2,8,9,16)(4,14,11,12)(5,15,10,13)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,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, 10050713);