#10159320

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18,22,23)(19,24,27,30)(20,25,31,28)(21,29,32,26)(33,34)(35,45)(36,42)(37,38)(39,43)(40,44)(41,48)(46,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,18,22,23)(19,24,27,30)(20,28,31,25)(21,26,32,29)(33,35,40,48)(34,41,44,45)(36,46,37,39)(38,47,42,43);(2,4)(7,9)(8,12)(15,16)(18,23)(19,27)(25,28)(26,29)(33,36,40,37)(34,42,44,38)(35,46,48,39)(41,43,45,47);(1,4,6,2)(3,9,11,7)(5,8,14,12)(10,15,13,16)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31)(33,37,40,36)(34,42,44,38)(35,39,48,46)(41,43,45,47);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,20)(18,25)(19,26)(21,24)(22,31)(23,28)(27,29)(30,32)(33,38,40,42)(34,37,44,36)(35,47,48,43)(39,45,46,41);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,21)(18,26)(19,28)(20,30)(22,32)(23,29)(24,31)(25,27)(33,39)(34,43)(35,36)(37,48)(38,41)(40,46)(42,45)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=g,e^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, 10066516);