#10196209

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4155 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=gh,e^2=g,f^2=i,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, 10103405);