#9003059

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

4274 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)(3,7,10,5)(4,8,12,15)(11,13,16,14)(17,18,23,26)(19,27,29,24)(20,30,31,21)(22,25,28,32)(33,34)(35,43)(36,41)(37,47)(38,39)(40,45)(42,48)(44,46);(1,3)(2,7)(4,11)(5,9)(6,10)(8,13)(12,16)(14,15)(17,19,23,29)(18,24,26,27)(20,25,31,32)(21,28,30,22)(34,39)(36,46)(40,43)(42,47);(3,10)(4,12)(5,7)(8,15)(20,31)(21,30)(22,28)(25,32)(33,35,38,45)(34,40,39,43)(36,47,46,42)(37,41,48,44);(1,4)(2,8)(3,11)(5,14)(6,12)(7,13)(9,15)(10,16)(17,20,23,31)(18,21,26,30)(19,25,29,32)(22,24,28,27)(33,35)(34,40)(36,47)(37,44)(38,45)(39,43)(41,48)(42,46);(1,5,6,7)(2,3,9,10)(4,13,12,14)(8,16,15,11)(17,21)(18,20)(19,28)(22,29)(23,30)(24,25)(26,31)(27,32)(33,36,38,46)(34,41,39,44)(35,42,45,47)(37,43,48,40);(1,3)(2,7)(4,11)(5,9)(6,10)(8,13)(12,16)(14,15)(17,22,23,28)(18,25,26,32)(19,21,29,30)(20,24,31,27)(33,37)(34,42)(35,44)(36,43)(38,48)(39,47)(40,46)(41,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=hi,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, 8910255);