#10040858

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3867 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:

(1,2,5,6)(3,7,10,9)(4,12,14,8)(11,16,13,15)(17,18)(19,24)(20,26)(21,27)(22,32)(23,25)(28,29)(30,31)(33,34,38,36)(35,40,43,39)(37,44,45,41)(42,46,48,47);(1,2,5,6)(3,9,10,7)(4,8,14,12)(11,16,13,15)(17,19,23,22)(18,24,25,32)(20,28,27,30)(21,31,26,29)(33,34)(35,39)(36,38)(37,41)(40,43)(42,46)(44,45)(47,48);(2,6)(3,10)(8,12)(11,13)(18,25)(20,27)(24,32)(28,30)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,20,23,27)(18,26,25,21)(19,28,22,30)(24,29,32,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,3,5,10)(2,7,6,9)(4,13,14,11)(8,16,12,15)(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);(1,4,5,14)(2,8,6,12)(3,11,10,13)(7,15,9,16)(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=gh,c^2=g,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9948054);