#9016617

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

4258 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:

(1,2,7,5)(3,8,6,13)(4,9,11,14)(10,15,12,16)(17,18,22,26)(19,21,25,23)(20,24,28,32)(27,30,29,31)(33,34)(35,43)(36,41)(37,47)(38,39)(40,45)(42,48)(44,46);(1,3,7,6)(2,8,5,13)(4,10,11,12)(9,15,14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(34,39)(36,46)(40,43)(42,47);(2,5)(3,6)(4,11)(15,16)(19,25)(20,28)(21,23)(24,32)(33,35,38,45)(34,40,39,43)(36,47,46,42)(37,41,48,44);(1,4)(2,9)(3,10)(5,14)(6,12)(7,11)(8,15)(13,16)(17,19,22,25)(18,23,26,21)(20,29,28,27)(24,30,32,31)(33,35)(34,40)(36,47)(37,44)(38,45)(39,43)(41,48)(42,46);(1,5,7,2)(3,8,6,13)(4,9,11,14)(10,16,12,15)(17,20,22,28)(18,24,26,32)(19,27,25,29)(21,30,23,31)(33,36,38,46)(34,41,39,44)(35,42,45,47)(37,43,48,40);(1,6,7,3)(2,8,5,13)(4,12,11,10)(9,15,14,16)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,29,32,27)(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=ghi,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, 8923813);