#9007451

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

4055 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8914647);