#8720370

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
prime radical	#46500	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,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, 8627566);