#10143509

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	#55684	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4160 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:

(1,2,6,7)(3,11,12,8)(4,9,13,16)(5,14,15,10)(17,18)(19,24)(20,26)(21,27)(22,32)(23,25)(28,29)(30,31)(33,34,38,36)(35,39,43,40)(37,44,45,41)(42,47,48,46);(2,7)(4,13)(8,11)(10,14)(17,19,23,22)(18,24,25,32)(20,28,27,30)(21,31,26,29)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(2,7)(3,12)(9,16)(10,14)(18,25)(20,27)(24,32)(28,30)(33,35,38,43)(34,40,36,39)(37,42,45,48)(41,47,44,46);(1,3)(2,8)(4,14)(5,9)(6,12)(7,11)(10,13)(15,16)(17,20,23,27)(18,26,25,21)(19,28,22,30)(24,29,32,31)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(1,4)(2,9)(3,10)(5,11)(6,13)(7,16)(8,15)(12,14)(17,21,23,26)(18,27,25,20)(19,29,22,31)(24,28,32,30)(33,36,38,34)(35,39,43,40)(37,41,45,44)(42,47,48,46);(1,5,6,15)(2,10,7,14)(3,9,12,16)(4,8,13,11)(17,22,23,19)(18,24,25,32)(20,30,27,28)(21,31,26,29)(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=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, 10050705);