#10166728

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

4060 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)(19,20)(21,29)(22,23)(24,26)(25,31)(27,30)(28,32)(33,34,38,39)(35,37,41,47)(36,40,45,48)(42,46,43,44);(4,11)(7,9)(12,16)(14,15)(18,23)(20,24)(25,29)(28,30)(33,35)(34,37)(36,43)(38,41)(39,47)(40,44)(42,45)(46,48);(2,3)(7,9)(8,10)(14,15)(18,23)(19,26)(25,29)(27,32)(34,39)(35,41)(40,48)(42,43);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,22,26)(18,24,23,20)(21,27,31,32)(25,30,29,28)(33,35)(34,37)(36,43)(38,41)(39,47)(40,44)(42,45)(46,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,24)(18,19,23,26)(21,30,31,28)(25,32,29,27)(33,36,38,45)(34,40,39,48)(35,42,41,43)(37,46,47,44);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,31)(18,25,23,29)(19,27,26,32)(20,28,24,30)(33,37)(34,35)(36,44)(38,47)(39,41)(40,43)(42,48)(45,46)

Polycyclic presentation:

2^9|a^2=gh,d^2=hi,e^2=gi,f^2=g,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, 10073924);