#8889343

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	#58174	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796539);