#8720772

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

4215 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,31)(29,30)(33,34,39,42)(35,40,37,41)(36,45,47,38)(43,44,46,48);(2,3)(7,9)(8,10)(15,16)(17,19,23,27)(18,24,25,22)(20,30,26,28)(21,29,32,31)(33,34)(35,40)(36,38)(37,41)(39,42)(43,48)(44,46)(45,47);(2,3)(7,9)(8,10)(15,16)(18,25)(19,27)(20,26)(29,31)(33,35,39,37)(34,40,42,41)(36,46,47,43)(38,44,45,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,23,26)(18,21,25,32)(19,28,27,30)(22,29,24,31)(33,36,39,47)(34,38,42,45)(35,43,37,46)(40,48,41,44);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21)(18,26)(19,29)(20,25)(22,30)(23,32)(24,28)(27,31)(33,37,39,35)(34,41,42,40)(36,46,47,43)(38,44,45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22,23,24)(18,27,25,19)(20,31,26,29)(21,28,32,30)(33,38)(34,36)(35,44)(37,48)(39,45)(40,46)(41,43)(42,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=gi,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, 8627968);