#8718263

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

4168 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625459);