#9017632

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

4236 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=hi,e^2=ghi,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, 8924828);