#8650333

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8557529);