#8712461

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	#55423	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4246 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)(19,23)(20,29)(21,26)(22,24)(25,30)(27,31)(28,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(2,3)(7,9)(8,10)(15,16)(17,19,22,27)(18,23,24,31)(20,28,30,26)(21,29,32,25)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(2,3)(7,9)(8,10)(15,16)(18,24)(19,27)(25,29)(26,28)(36,45)(40,44)(42,48)(46,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19)(18,23)(20,28)(21,25)(22,27)(24,31)(26,30)(29,32)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20,22,30)(18,25,24,29)(19,28,27,26)(21,31,32,23)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21)(18,26)(19,29)(20,23)(22,32)(24,28)(25,27)(30,31)(33,37,38,34)(35,41,43,39)(36,40,45,44)(42,46,48,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=g,e^2=h,f^2=g,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, 8619657);