#9143390

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

4016 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9050586);