#9017310

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

4225 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)(3,5)(4,9)(6,16)(7,10)(8,14)(11,13)(12,15)(17,18)(19,21)(20,29)(22,30)(23,25)(24,28)(26,27)(31,32)(34,36)(39,43)(40,44)(47,48);(1,3,7,6)(2,5,10,16)(4,12,14,11)(8,13,9,15)(17,19,23,28)(18,24,25,21)(20,26,30,32)(22,27,29,31)(33,34,38,36)(35,39,42,43)(37,40,46,44)(41,47,45,48);(1,4,7,14)(2,8,10,9)(3,11,6,12)(5,15,16,13)(17,20,23,30)(18,22,25,29)(19,26,28,32)(21,31,24,27)(33,35,38,42)(34,39,36,43)(37,45,46,41)(40,48,44,47);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,10)(13,16)(17,20)(18,22)(19,26)(21,31)(23,30)(24,27)(25,29)(28,32)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,5)(2,3)(4,13)(6,10)(7,16)(8,12)(9,11)(14,15)(17,21,23,24)(18,19,25,28)(20,27,30,31)(22,32,29,26)(33,35)(34,39)(36,43)(37,41)(38,42)(40,47)(44,48)(45,46);(1,6,7,3)(2,5,10,16)(4,11,14,12)(8,13,9,15)(17,22)(18,20)(19,27)(21,32)(23,29)(24,26)(25,30)(28,31)(33,37,38,46)(34,40,36,44)(35,41,42,45)(39,47,43,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=ghi,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^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924506);