#9206770

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

4185 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113966);