#9020263

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

4081 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=h,f^2=i,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, 8927459);