#8994417

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

4161 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=i,e^2=h,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, 8901613);