#10195478

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	#42712	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(18,23)(19,26)(25,32)(27,28)(33,34,40,43)(35,44,47,41)(36,48,38,39)(37,45,46,42);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,18,22,23)(19,24,26,21)(20,25,30,32)(27,29,28,31)(33,35,40,47)(34,41,43,44)(36,45,38,42)(37,39,46,48);(2,3)(7,9)(8,10)(14,15)(33,36,40,38)(34,39,43,48)(35,45,47,42)(37,41,46,44);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,24)(20,28)(21,23)(22,26)(25,31)(27,30)(29,32)(33,37,40,46)(34,42,43,45)(35,39,47,48)(36,44,38,41);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,38,40,36)(34,39,43,48)(35,45,47,42)(37,44,46,41);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,28,32,27)(33,39)(34,38)(35,46)(36,43)(37,47)(40,48)(41,42)(44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=h,e^2=g,f^2=ghi,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, 10102674);