#9481415

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

4049 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)(3,7)(4,9)(5,13)(6,8)(10,14)(11,12)(15,16)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,31)(30,32)(35,41)(39,46)(42,44)(47,48);(1,3)(2,7)(4,8)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(2,8)(3,9)(5,14)(12,15)(21,29)(24,31)(26,32)(28,30)(34,38)(39,46)(40,43)(47,48);(1,3)(2,4)(5,11)(6,9)(7,8)(10,12)(13,15)(14,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31)(33,35,37,41)(34,39,38,46)(36,44,45,42)(40,48,43,47);(1,4)(2,9)(3,8)(5,12)(6,7)(10,16)(11,13)(14,15)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)

Polycyclic presentation:

2^9|b^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388611);