#9333331

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

4188 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9240527);