#9164019

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

4209 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)(3,8)(4,12)(5,9)(6,10)(7,13)(11,15)(14,16)(17,18,23,21)(19,24,27,29)(20,25,31,30)(22,26,32,28)(34,38)(35,41)(40,43)(42,44);(1,3)(2,5)(4,11)(6,9)(7,15)(8,10)(12,16)(13,14)(17,19,23,27)(18,24,21,29)(20,28,31,26)(22,30,32,25)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(4,13)(7,12)(11,14)(15,16)(19,27)(20,31)(24,29)(25,30)(33,35,37,41)(34,39,38,46)(36,42,45,44)(40,47,43,48);(1,4,6,13)(2,7,10,12)(3,11,9,14)(5,15,8,16)(17,20)(18,25)(19,28)(21,30)(22,24)(23,31)(26,27)(29,32)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,3)(2,8)(4,14)(5,10)(6,9)(7,15)(11,13)(12,16)(17,21,23,18)(19,29,27,24)(20,25,31,30)(22,26,32,28)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48);(1,5,6,8)(2,9,10,3)(4,15,13,16)(7,14,12,11)(17,22)(18,26)(19,30)(20,24)(21,28)(23,32)(25,27)(29,31)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9071215);