#9160755

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

4189 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)(4,10)(5,11)(6,14)(7,9)(12,15)(13,16)(17,18)(19,26)(20,28)(21,23)(22,25)(24,29)(27,31)(30,32)(33,34,40,43)(35,44,47,41)(36,48,38,39)(37,45,46,42);(1,3,7,12)(2,8,9,15)(4,13,14,5)(6,11,10,16)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32)(33,35,40,47)(34,41,43,44)(36,45,38,42)(37,39,46,48);(2,9)(3,12)(4,14)(11,16)(20,29)(24,28)(27,30)(31,32)(33,36,40,38)(34,39,43,48)(35,45,47,42)(37,41,46,44);(1,4)(2,10)(3,13)(5,12)(6,9)(7,14)(8,16)(11,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32)(33,37,40,46)(34,42,43,45)(35,39,47,48)(36,44,38,41);(1,5,7,13)(2,11,9,16)(3,14,12,4)(6,15,10,8)(17,18)(19,23)(20,28)(21,26)(22,25)(24,29)(27,32)(30,31)(33,38,40,36)(34,39,43,48)(35,45,47,42)(37,44,46,41);(1,6,7,10)(2,4,9,14)(3,11,12,16)(5,8,13,15)(17,21,22,19)(18,23,25,26)(20,30,29,27)(24,31,28,32)(33,39)(34,38)(35,46)(36,43)(37,47)(40,48)(41,42)(44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=hi,e^2=h,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, 9067951);