#10143414

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	#48573	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3928 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,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, 10050610);