#9017589

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

4226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=hi,e^2=hi,f^2=h,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, 8924785);