#10196216

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

4210 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)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34)(35,45)(36,39)(37,46)(38,42)(40,44)(41,48)(43,47);(4,11)(7,9)(12,16)(14,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35)(34,41)(36,46)(37,42)(38,43)(39,47)(40,48)(44,45);(2,3)(7,9)(8,10)(14,15)(18,19)(23,25)(24,26)(30,31)(33,36,40,38)(34,42,44,39)(35,46,48,43)(37,45,47,41);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,37,40,47)(34,43,44,46)(35,42,48,39)(36,41,38,45);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,38,40,36)(34,42,44,39)(35,46,48,43)(37,41,47,45);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,39,40,42)(34,38,44,36)(35,47,48,37)(41,43,45,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=h,e^2=g,f^2=gi,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, 10103412);