#10168447

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=i,e^2=ghi,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, 10075643);