#9685616

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

4216 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)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,22)(19,25,20,24)(21,26,29,30)(27,32,28,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,20)(18,24,22,25)(21,27,29,28)(26,31,30,32)(35,41)(38,45)(42,47)(46,48);(2,3)(7,9)(8,10)(14,15)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,19)(18,25,22,24)(21,28,29,27)(26,32,30,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,18)(19,25,20,24)(21,26,29,30)(27,31,28,32)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

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, 9592812);