#10143757

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,22,26)(19,21,25,23)(20,24,30,28)(27,31,32,29)(33,34,40,44)(35,41,46,48)(36,42,37,39)(38,43,45,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,29,28,31)(33,35,40,46)(34,41,44,48)(36,47,37,43)(38,39,45,42);(2,6)(7,9)(8,12)(15,16)(33,36,40,37)(34,42,44,39)(35,43,46,47)(38,41,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,31)(29,32)(33,37,40,36)(34,39,44,42)(35,43,46,47)(38,41,45,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,38)(34,43)(35,39)(36,41)(37,48)(40,45)(42,46)(44,47);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,39)(34,36)(35,45)(37,44)(38,46)(40,42)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=hi,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, 10050953);