#10143668

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	#38590	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=ghi,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, 10050864);